(CNN) -- A French woman seized in northern Kenya was taken to Somalia by her captors after a sea battle to rescue her failed, Kenyan government officials said.
Gunmen snatched the woman early Saturday from her holiday home in Manda Island, close to where a Briton was abducted and her husband killed last month.
They were "10 heavily armed Somali bandits suspected to be al-Shabaab operatives" from Ras Kamboni, across the border in Somalia, Kenyan security minister George Saitoti said in a statement.
Security forces pursued the men as they raced in a high-speed boat toward the border, but despite the Kenya navy injuring several abductors in a shoot-out, they managed to reach Ras Kamboni, according to Saitoti.
"In the meantime, every effort is being made to rescue the victim," he said.
Andrew Mwangura, a piracy expert and editor of Somalia Report, said his sources in Somalia told him the French hostage has been taken deeper into the country.
She was taken to a town in the north, Mwangura said, which makes her rescue more complicated in an area run by the al-Shabaab.
French authorities are not releasing the woman's name, said Eric Bosc, a spokesman for the French Foreign Affairs Ministry.
She is in her 60s and lives in Kenya about six months a year, and French authorities are doing everything they can to free her, the spokesman said.
Kenyan officials and an acquaintance said the woman uses a wheelchair and is not in good health.
It is the second kidnapping near the popular tourist town of Lamu in a month. Gunmen fatally shot a British man and kidnapped his wife, Judith Tebbutt, from a safari lodge near the town last month.
David Tebbutt was killed in the attack when he resisted, according to Kenyan police.
The U.K. Foreign and Commonwealth Office issued revised travel advice to its citizens traveling near the Somali border.
"We advise against all but essential travel to coastal areas within 150 km of the Somali border, following two attacks by armed gangs in small boats against beach resorts in the Lamu area on 11 September and 1 October 2011," the statement said.
Its previous advice cautioned against travel within 60 kilometers (37 miles) of the border.
Security analysts have said Judith Tebbutt is being held by pirates in a remote corner of Somalia. British government officials have asked journalists not to reveal her exact location to avoid abduction attempts from rival gangs.
The French embassy in Nairobi urged "extreme caution" for people visiting Lamu and the surrounding area.
Manda Island is made up of luxury homes and some small hotels. It is directly across a narrow channel from Shella Beach, one of the most popular -- and long considered safest -- tourist spots in Kenya, on Lamu Island. The channel runs straight from the open ocean.
It is off-season in the area, an is frequented by Europeans and other expatriates living in Kenya.
Hadija Ernst, a resident of the area and editor of the local magazine Chonju, said the abductions have put local residents and tourists on edge.
"The people are actually leaving, the tourists that are here are leaving because of the reports that we're getting," she said.
Ernst, who knows the kidnapped woman, said she is a "kind and gentle person."
"She's very interested in Swahili culture, the culture of the island here," she said.
Security in the area should have been improved following the first kidnapping, Ernst said. She has not seen any signs of increased security in the area and worries that the tourist economy in the area will suffer.
"So we are all very concerned and we want to see the government take this matter very seriously," she said.
Saitoti called the abduction a serious violation of Kenya's territorial integrity.
CNN's Zain Verjee and David McKenzie contributed to this report.