David Tebbutt, 58, and his wife Judith, 56, were on holiday at the remote Kiwayu Safari Village, close to the Kenyan border with Somalia, when the attack happened during the early hours of yesterday morning.
There are suspicions that the gang was from Somalia and used a speedboat to get away from the isolated island resort, which has played host to artist Tracey Emin, actress Imelda Staunton and reportedly Sir Mick Jagger.
The Foreign Office has confirmed that two British nationals were attacked overnight at a beach resort north of Lamu, near the Kenya-Somalia border.
A spokesman said: "One was killed and another kidnapped. We are working closely with the Kenyan authorities to establish further details.
"We have deployed a consular team from our High Commission in Nairobi and are offering all possible support to the family of those involved. Our thoughts are with them at this difficult time.
"We are working to secure the safe and swift release of the British national who has been kidnapped and ask those involved to show compassion and release the individual immediately."
The Foreign Office urges against "all but essential travel to within 30km of Kenya's border with Somalia".
Its website says: "There have been previous attacks by Somali militia into Kenya. Three aid workers were kidnapped in July 2009, and two western nuns in November 2008."
It also warns against piracy, referring to the kidnapping of two British nationals in October 2009 as they sailed from the Seychelles to Tanzania in notoriously dangerous waters.
Retired couple Paul and Rachel Chandler, from Tunbridge Wells in Kent, spent 388 days in capitivity until they were released last November after a ransom of up to one million dollars (£620,000) was paid.
The Foreign Office website also warns: "If you visit Lamu Island, do so by air if possible. This is for security reasons and also because of the bad road conditions.
"Buses and other vehicles on the road to Lamu have been attacked by armed robbers in the past and overland travel from Lamu to Malindi should only be undertaken in an armed police convoy."
As the search continued for Mrs Tebbutt today, the Kenyan military was said to have joined the hunt.
Eric Kiraithe said Kenyan authorities were leaving no stone unturned and police and military were combing the area.
Officials have said the Tebbutts were attacked on the first night of their stay at the resort, with reports suggesting that bandits broke into their accommodation.
Police Commissioner Mathew Iteere said yesterday that the couple had come from visiting the Masai Mara reserve and were the resort's only guests.
He said bandits could gain access to the couple's accommodation, which had a piece of cloth as a door, "so easily".
He told a press conference it was possible that Mr Tebbutt had "resisted", which may have been why he was shot.
"Maybe they wanted to take the two but he resisted, that's why he was shot," he said.
He said that if the attackers were hoping for a ransom for Mrs Tebbutt, it was likely they would get in contact.
There has been speculation that the kidnappers could be from an al Qaida-linked insurgent group called al Shabab, which holds much of southern Somalia.
Mr Tebbutt, believed to be from Bishop's Stortford, Hertfordshire, worked for publisher Faber & Faber and was a member of the Book Trade Charity, which offers support and grants to those in the book trade.
Chief executive David Hicks, who met him more than 10 years ago, said the best way to describe Mr Tebbutt was "all the dedicated words you want".
"He was a lovely chap, he was on the grants committee. He was a very caring person and very concerned about the people that we were supporting financially."
He said he knew Mr Tebbutt, who was also a member of literary dinner club the Society of Bookmen, had lived and worked in Africa previously, although he did not know the details, but he did not know his wife.
The Kiwayu resort's website states it takes "security and safety very seriously".
It says: "Our relationship with the local community, its fishermen and the local authorities is positive and mutually beneficial.
"We regularly review our security and safety to ensure it is both comprehensive and current."
Consisting of 18 luxury cottages spread along a private beach, the resort is popular among backpackers and celebrities alike, including Emin and Staunton, who have both written of their experiences.
The resort's online brochure also says: "There are few places in the world which can boast such seclusion.
"We are visited only by fishermen, the dhows that still come to collect water and the occasional sports fisherman in search of the magnificent marlin, which are then tagged and released back into the sea."
It currently charges 445 US dollars (around £278) per adult per night to stay in its beach bungalows, which boast locally carved furniture and hammocks.
PA
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