Daddy's girl: Mohamed al-Fayed's daughter Jasmine is gaining a reputation as a hot young designer

Finding the Jasmine di Milo boutique in Harrods is not easy. "Jasmine who?" replies a bemused sales assistant. "Is it lingerie?" asks another blankly. Perhaps using the designer's real surname – Fayed – will help to clarify matters. Sure enough, now, it seems, we are talking. Cue much flapping and the hasty production of a floor plan.

It's a momentary insight into the power of a family name – in this case, one that is fraught with colourful associations. It is not difficult to imagine why Jasmine al-Fayed, eldest daughter of Harrods' controversial owner Mohamed, might have wanted to ditch hers when christening her womenswear label.

But if the 28-year-old has nominally distanced herself from her famous lineage, in practice she has remained close to her father's empire. In 2003 she opened her first store within the Harrods fashion department amid inevitable accusations of nepotism of the sort that dogged the early years of Stella McCartney's design career. When she showed her first collection at Paris Fashion Week at the Fayed-owned Ritz in 2006, one British newspaper observed caustically that it was "proof that daddy's money really can't buy you style. But it can buy you a fashion show in Paris".

Two years on, however, and the tide of popular opinion slowly seems to be turning in Fayed's favour. Her now-signature sculptural cocktail dresses and fluid evening gowns are gracing red carpets around the world with increasing regularity, courtesy of A-listers such as Thandie Newton, Sienna Miller and Anne Hathaway. Stocked in more than 44 countries, the label is proving a hit with a young, international crowd who want – and can afford – high-end fashion that is glamorous but youthful and fun. Heck, even a few journalists are saying they like it.

Sitting in her studio, in the shadow of London's most famous department store, Jasmine al-Fayed is not the Knightsbridge princess one might expect. There are the tattoos, for one thing. Waif-like in her standard day ' uniform of low-key skinny jeans and a grey T-shirt, one slender forearm reveals a large tiger's head crowned with the words "GOD IS LOVE". On the other is a pair of wings. (What did her dad make of those? "Oh, he just did this little Muttley laugh. A wry smile," she chuckles. "He knew if he'd verbally indulged me I'd have had a whole sleeve done.") With her Aviators perched in her glossy hair and feline black eyeliner, her look is west London rock chick. Polished, with an edge.

If her face is unfamiliar, that's because Fayed has chosen to keep it that way. Unlike the rest of London's celebrity offspring, she is rarely photographed at parties. She is rarely photographed at all, in fact, preferring a quieter life with boyfriend Noah, a musician, and their two-year-old daughter. In a telling error, next to her entry in a recent "Hot thirty under 30" list, women's magazine Grazia printed a picture of her younger sister, the 22-year-old Camilla, who is spotted more often on the pages of Vogue and Tatler.

So far, Fayed has given only a handful of interviews to the British media, an unusual decision for anyone with a product to push, no matter who your backers are. There are to be no questions about "the brother", by which I assume her PRs are referring to her late half-brother Dodi, the product of her father's first marriage to Samira Khashoggi, who died in the car crash that killed Diana, Princess of Wales 11 years ago. Before I have a chance to switch on my own tape recorder, a second, higher-tech, digital device is produced by one of Fayed's assistants, putting my old Dictaphone to shame.

It's a surprise as much as a relief, then, to find that Fayed herself is relaxed, good-humoured and often disarmingly frank. I tell her I had presumed her to be more..."Media-shy?" she offers, shaking her head. "No, I just do what I need to do for the business and no more. I don't want to be a human tourist attraction. I was the second-eldest when my brother passed away and I think it did impact me the most. Seeing that kind of furore around that time put me off for life. I cherish my personal life. It impacted me insofar as I knew that I was more the creative recluse rather than the big public..." She trails off. "It confirmed my personal instinct."

Nevertheless, Fayed does not play the victim card ("I'm not a wallflower," she insists) and there is a quiet steeliness about her that one imagines to be the inevitable by-product of years of witnessing the people closest to you cast as characters in all manner of unseemly dramas. Not surprisingly, her admiration for, and loyalty to, her father is absolute, and she talks fondly about a childhood spent on their rural Surrey estate or accompanying her father to work at Harrods: "My dad loved having us around so he would take us everywhere. So if he did a store-walk, we'd do a store-walk."

Does she still find it upsetting to see him portrayed negatively in the press? "I think...," she hesitates. "I think it doesn't upset me because I think he's grossly misunderstood and it also amuses me to see how quickly people jump, especially when they have establishment views to preserve and protect... Whoever has been outspoken in a cause or who has had a massive bone to pick either gets ridiculed or demeaned. He's been a fantastic inspiration in one way, in that he's just one resilient armadillo."

She also appears to echo some of her father's more contentious claims that her brother's death was the result of a secret plot orchestrated by British security services: "He really believes and he has reason to believe. What obviously can't be put out into the press [about Dodi's death] we know as a family, so living with that gives you some peace. I think there are so many things in the public mind that haven't been put to bed, but they have for us."

This ability to dissociate herself and her experiences from their various public representations has surely been the key to the success of her own enterprise, for most would have baulked at the idea of laying themselves and their work bare to a media that was hardly disposed to welcome her as a serious young designer.

Although Fayed won places at London College of Fashion and Central St Martins (going under the maiden name of her mother, Heini Wathen, a former Miss Finland who has never been one for the limelight either), she opted not to complete her degree. Instead, she served a one-year apprenticeship in the sewing workrooms at Harrods and took several short courses "to fill the gaps in my knowledge". With backing from her father, she launched a capsule collection that she sold alongside a hand-picked selection of other nascent labels in her Jasmine di Milo mini-boutique within the department store. The name for her brand, she explains, was inspired by the Greek sculpture: "I love the story of Venus de Milo – the idea that there is no creator. She's the ultimate mystery."

I wonder aloud whether she feels she damaged her credibility by choosing to launch herself in her father's store, in the same way that Stella McCartney's decision to enlist the help of her supermodel friends at her graduation show did little to endear her to her peers. In both cases, to have such resources at one's disposal has proved a double-edged sword.

Fayed squares up to such charges unapologetically. "You could look at it in another way," she says. "In that it would almost be strange to deny where I am from. I didn't overplay it or underplay it and I sleep soundly on my pillow knowing that everything I did was with integrity and good intention." Her occasional hyperbolic moral rhetoric is reminiscent of her father in full crusade mode, although she delivers it with a calm directness that seems artless.

Good intentions aside, the harsh reviews of Fayed's early collections must have been discouraging. "I read a couple of things," she shrugs. "I don't alienate myself from what people are saying. You just see what's out there and you take it or leave it... I didn't let it stop me discovering for myself what I could and couldn't do... I just put my head down and got on with my work, and the pieces speak for themselves."

While Fayed's first creations indulged her personal penchant for glam-punk and gothic styles, she has since streamlined things, simplifying shapes and embellishment. Mini-dresses are structured and contemporary, floor-length goddess gowns have red carpet written all over them and there are plenty of wearable, cleanly tailored separates. "I've always liked the rock'n'roll undercurrent," says Fayed, who confesses that her style icon as a teenager in the 1990s was Aerosmith's Steven Tyler. "Now maybe it's slightly older. It's definitely simplified. I have a broader reach with my designs."

If Fayed has a creative vision for the brand, her commercial proposition is equally important: "The balance that I'm trying to strike right now is sort of demi-couture, one-of-a-kind but you're not paying £10,000 for it. It's part of my responsibility to a) make it affordable, just reasonable for the quality and the materials that I use, and b) to start doing smaller bursts so that there's always something new dropping." It's an admirable aspiration, particularly when you consider that Fayed would have little trouble getting high-spending Harrods customers to part with considerably more cash than the average £400 for a cocktail dress.

With the label in its sixth season, Fayed's grit is paying off. The Jasmine di Milo celebrity fanbase appears to grow by the day, and the press is getting better too. One fashion editor I speak to suggests this is a combination of the evolving maturity of her designs and some very canny PR gifting of dresses to the right stars. Fayed flatly rejects the latter suspicion: "I don't do the whole giving thousands of freebies away and hunting people down thing. If they find it and they like it, that's how I forge a relationship."

In her characteristically offbeat way, she reveals that the celebrity she has most enjoyed dressing is not one of the bevy of Hollywood starlets who have worn her designs, but the singer Prince, who saw the outfits she had provided for his backing dancers and asked her to make him a one-off suit. For the moment, however, menswear is not on the agenda. Rather, the next step for Jasmine di Milo will be childrenswear, she explains, gesturing to a rack of sweet, old-fashioned silk baby dresses that she has been working on with her daughter in mind.

Does she have any designs on her father's wardrobe? "I've tried and failed but he is his own creature," she says fondly. "It suits his personality. He just has that funk factor that only he can pull off."

For now, she is more concerned with the final preparations for her Paris show next week, which she confides will be an opportunity to do what she loves best: get out of London. "I'm not built for the metropolis," she explains. "I love to get away – next stop is Canada. I'm an earthy girl. I like my feet in the elements." It is perhaps a surprising statement from a designer of chic cocktail dresses. And from a billionaire heiress. But then, surprising people is proving to be what Jasmine al-Fayed does best.

For a full stockist list, visit www.jasminedimilo.com