If it's an obsession that has spilt over into adulthood – "I'm going to call my daughter 'Alice', if I ever have one," she states – it's more than just a childish hangover. The red-haired Gillan, who at 5ft 10in is far taller than she appears on screen, admits she thought of Alice when she was working on her Doctor Who character, Amy Pond.
"I remember thinking: 'What's her favourite film?' And I thought it could be Alice in Wonderland."
It seems an apt choice, given Amy – dragged from her mundane existence to travel time and space with Matt Smith's new Doctor – gets to "explore" her very own Wonderland.
The same could be said for Gillan, who has gone from bit-part actress and part-time model to landing one of the most coveted roles on television. Her arrival on the show couldn't have been more arresting, with Amy, a kiss-o-gram by trade, spending the first episode dressed in a policewoman's uniform. The tabloids had a field day, renaming the show 'Doctor Phew'.
Even Smith weighed in, claiming Gillan was "categorically the sexiest companion ever", knocking such recent co-stars as Catherine Tate and Billie Piper for six. But Gillan seems rather nonplussed by the instant sex symbol status. "It doesn't feel like a massive achievement," she says.
That first episode broke the record for the biggest audience ever on BBC America when it aired in the States.
"We did this one screening in New York where people camped out overnight to get a seat," she recalls, breathlessly. She can even ponder the fact that she's now on Steven Spielberg's radar. A long-term fan of the show, the director has just worked with Doctor Who's new series producer Steven Moffat, who has co-written Spielberg's forthcoming Tintin film, The Secret of the Unicorn. "It's mad to think that he's a fan of it," she says. "Spielberg apparently said to Steven: 'The world is a better place with Doctor Who in it.'"
Gillan seems almost embarrassed when I point out that being in Spielberg's mind is no bad thing. After all, her film experience so far has consisted of playing Young Girl in Bus Station in Richard Jobson's frenetic 2008 thriller New Town Killers.
Next week sees a bigger appearance in first-time director Colm McCarthy's Outcast, a British horror film co-starring James Nesbitt and Red Road's Kate Dickie, that sets out to blend the monster movie genre with Polanski-style terror. "It's like a dark, scary fairy-tale," she explains. "It's all about Celtic mythology."
Admittedly, the film – which sees Nesbitt's "tracker" use black magic to trap his human prey – casts Gillan as a sidekick once again, playing best friend to the film's mischievous lead girl Petronella (Hanna Stanbridge), who becomes embroiled with one of Nesbitt's targets. But what it does do is give Gillan the chance to establish her film credentials before playing Amy Pond pigeonholes her in a Tardis-shaped box for good. While the likes of Tate and Piper were able to shake off the Doctor Who shackles, perhaps due to their pre-Who careers in comedy and music, the 23 year-old Gillan does not have that luxury.
Though attached to David Baddiel's US-set time-travel comedy Romeo and Brittany, for the moment, she can't think beyond the next few months.
With her first season – the fifth, since the show returned to our screens in 2005 – now safely out on DVD, she is filming the next series, which as usual will be unveiled at Easter. "I think a lot's going to happen to Amy!" she teases. "A lot!"
First, though, is another Doctor Who tradition – the Christmas Special. This year's episode is a twist on Charles Dickens' A Christmas Carol. So will she sit round with the family on Christmas Day and watch it, like everyone else? "Probably," she stammers. "I guess so. Is that weird?" It's not, I assure her.
Gillan's immediate family is a small one. Raised an only child by father John and mum Marie, she admits she always hankered for a brother or sister when she was young, a feeling that hasn't dissipated. "It freaks me out – and this sounds really morbid – that when your parents pass away there will be no-one there that feels the same as you." Her Alice in Wonderland obsession aside, her first real interest was music, begun after she started learning the piano when she was just 7. "My Dad's a massive music fan, so I was always subjected to Ella Fitzgerald, Nina Simone, Elvis Presley and Frank Sinatra," she says. "That was my first love."
While she eventually switched her attentions to acting in local plays, what she didn't like was education.
"I just remember really hating being in school," she says. "I used to sit in fashion class in school – where we had to make stuff – and I'd sneak off and go and look on the computer for auditions on The Stage website. And I'd imagine going down to London to do all that, rather than be stuck in school."
Another example of her vivid imagination, in her own eyes Gillan was a bit of an oddity in her teenage years.
"I was that weird, long, ginger girl at school," she says. "I wasn't horrifically bullied. There was name-calling but nothing awful." Moving to London when she was 17, and enrolling in the Italia Conti drama school, Gillan left after a few months, when she was offered a role in one episode of Scottish detective show Rebus. "I was really restless, because I wanted to be acting," she reasons. "But I was thinking: 'Was it the right thing to leave drama school for this one episode?' Whereas I could've stayed there for three years and got a degree. But I don't know. I'm just like that. I just left randomly. I was longing to get back into acting, but it was quite difficult."
Gillan wound up working in a pub, in south London, to make ends meet.
"I think that was really important for me," she says. "So I can appreciate this all the more."
She started working as a model, even taking part in the launch party for Dainty Doll, a make-up range endorsed by that other famous redhead, Girls Aloud star Nicola Roberts.
In the show, each model took on a character. Hers was Premiere Girl – a red-carpet strutting glamour-girl. All very apt, given where Gillan seems heading, but when I ask her if she has any desire to re-ignite her modelling career, she seems ambivalent.
"It's not something I'm desperate to do. But it could be fun. What girl wouldn't want to do that?" What about echoing Roberts and starting her own fashion or cosmetics line? "No, I wouldn't do that," she says, firmly.
Already, there have been the obligatory paparazzi shots of Gillan leaving a party looking "worse for wear". "Oh," she says, deflating the non-event with a pin-like jab of her tongue. Does it bother her? "I haven't been on the scene long enough to be getting all irritated about it. It's part of the job. I've got a personal life that nobody knows about." Well, that's not entirely true.
Gillan has been shot with her 24-year-old boyfriend, photographer Patrick Green, more than once. But she refuses to get drawn on the topic.
Naturally, there was initial speculation linking her to Matt Smith, long-since dismissed, when it was revealed that the actor is dating model Daisy Lowe. Gillan's affection for her Doctor seems purely platonic. While Smith might dub her sexy, she refrains from going down that route.
"He's much more other-worldly than the other Doctors," she says. She recalls her second audition when she first saw Smith in character.
"He just looked mental. He was going so mental that he was spitting. I was thinking: 'He looks like a mad scientist!'"
Outcast opens on December 10th. Volumes 1 to 3 of Doctor Who Series 5 are now available to buy on DVD. The Doctor Who Christmas Special will screen on BBC1 on Christmas Day.
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