An attacker fatally stabbed a female police administrative worker on Friday at the entrance to a police station in Rambouillet, a middle-class commuter town outside Paris.
Paris region president Valerie Pecresse said that terrorism could not be ruled out as a motive. Another official said separately that France's anti-terror prosecutor had not yet decided to lead the investigation.
The woman was knifed in the throat, two security sources said.
Assailant Shot Dead
"France has lost one its everyday heroines in a barbaric gesture of infinite cowardice," Prime Minister Jean Castex said on the scene of the attack.
The assailant was shot dead by police officers, one of the two security sources said.
The attacker was of Tunisian nationality and residing in France on legitimate papers, and there was no immediate indication he had cried any Islamist or Islamic slogans, a security official said.
He was not previously known to France's security agencies, a third security source added.
France has suffered a wave of attacks by Islamist militants or Islamist-inspired individuals in recent years that have killed about 250 people.
Friday's attack was six months after an Islamist teenager beheaded a school teacher in Conflans, another Paris satellite town.
Tackling religious extremism, domestic security and notions of French identity are likely to be important issues in next year's presidential election.