Murugesu Jayaganesh Pakeerathy, 41, was arrested on Monday when she was at the airport with her eight-year-old daughter, a French national, and initially detained for 72 hours pending investigations by an anti-terrorism unit, said police spokesman Ajith Rohana.
"We have now got a detention order under the Prevention of Terrorism Act which means we can hold her for a period of 90 days," Rohana told AFP.
"We are hopeful of completing investigations very soon."
The French embassy was providing consular assistance but gave no further details.
Police said the mother was believed to have been a leader of the Sea Tigers, the naval unit of the Liberation Tigers of Tamil Eelam (LTTE). She had been an active combatant between 1997 and 2000, Rohana said.
She had left the island in 2005, four years before the LTTE was crushed by troops in the bloody finale to a 37-year separatist conflict.
Her arrest comes despite an announcement by the new government of President Maithripala Sirisena that it would launch a fresh ethnic reconciliation process with the help of the international community.
