A gang of 10 armed men seized Marie Dedieu, 66, from Manda Island in the Lamu archipelago, and fled by sea, fighting off an attempt by Kenya's navy to stop them.
The kidnap, the second in the area in less than a month, has dealt a further blow to Kenya's tourist trade, with many hotels on Manda closed or empty.
Kenyan officials say they have sent mediators to Somalia to establish contact with the abductors.
The officials say they suspect Somali Islamist Al Shabaab insurgents carried out the abduction, but sources in Somalia dismiss that theory.
Stefano Moccia, the owner of the Majlis hotel on Manda told AFP police chief Mathew Iteere had spent the night in his hotel and told him the police planned to "build two police posts on Manda and have a helicopter stationed there."
An AFP journalist on the island saw one post being built from corrugated iron.
A man from Manda Island, identified by police as an employee of Dedieu, has been detained for questioning.
"The man we have in custody was working at the womans home and he is assisting us with the investigation," said a police source who asked not to be identified.
However residents of Manda Island said the man arrested was in fact the guard on a plot of land next to Dedieu's house.
A Somali resident of Ras Kamboni, where Dedieu was initially taken by her captors, confirmed the kidnappers numbered around 10.
"A car was waiting for her when she was brought in and she was taken to Odow, nine kilometres (five miles) east of Ras Kamboni."
Later Saturday or Sunday she was transferred, again by car, to the Kismayo area, the source said, adding Dedieu was not being held in Kismayo itself, which is under Al Shabaab control, but rather a little way out of town.
Adbul Alim, a close friend of Dedieu's, who said she is also fighting cancer, said she is on one type of medication she has to take every four hours.
This latest kidnapping sparked an immediate exodus of worried visitors, but Tourism Minister Najib Balala pleaded for calm.
"An emotional reaction is not going to help to solve the issue," he told reporters on Monday. "The minute you close down, you create a panic."
However, police chief Iteere warned ensuring proper security in the region's resorts is no easy task.
"There are 252 hotels and a bit over 2,000 villas and cottages on the coast, it is almost impossible to have static security for all of them," he said.
A British tourist, Judith Tebbutt, was seized to the north of Lamu and taken to Somalia on September 11th by an armed gang who killed her husband.
She is believed to have been sold to pirates now holding her in central Somalia.
The security forces had insisted that the first attack, for which two Kenyans have been charged, was an isolated incident.
The latest kidnapping prompted France and Britain to warn travellers to avoid not only Somalia but the nearby Kenyan coastline as well, to the dismay of Lamu's hotel and tour operators.
The tourism sector, a key component of Kenya's economy along with tea and horticulture, already took a hammering in 2008 after post election violence that left some 1,300 dead and displaced tens of thousands.