The woman was also injured in the neck but has since left hospital.
The attacker, who was carrying religious materials including a Quran during the attack, shouted "Allahu Akbar", or "God is great", according to a witness, the prosecutor's spokesman said.
"An act of terrorism can not be ruled out just because a person has psychiatric problems, but our investigation is still ongoing", French public prosecutor Laure Beccuau told a news conference on Saturday.
The police have opened an investigation of murder and attempted murder.
France remains on high alert after being hit by a string of attacks by jihadist extremists since 2015, with more than 250 people killed in total.
Police eventually killed all three attackers but it was the beginning of the wave of strikes on French territory.
Counter-terror prosecutors are investigating a knife attack carried out near Paris on Friday, the French Ministry of Justice said.
French junior interior minister Laurent Nunez, visiting the scene, praised the police response to the attack, labeling it "extremely courageous".
The attacker's family was co-operating fully with police, she added.
Nathan C converted to Islam in mid-2017 and is believed to have suffered serious psychiatric problems since he was child, with several spells in hospital.