Prosecutor Arnaud Laraize said earlier that the suspect was an Oleron resident and initially reported that 10 people had been injured.
Laraize said the man – already known to the police for alleged involvement in petty crimes – cried “God is the greatest” in Arabic when arrested.
The expression is a key refrain in Islam, but also used by militants carrying out attacks.
Police used a Taser to arrest the suspect as he set fire to his car, Nunez said.
The prosecutor said the man ran over people on a road between Dolus d’Oleron and Saint-Pierre d’Oleron.
The mayor of Saint-Pierre d’Oleron told local journalists the man lived in a mobile home in his town and was “someone who lived alone, had a very isolated life”.
An MP from the anti-immigration National Rally (RN) party said the woman who was gravely wounded was his aide and had been out for a morning jog.
RN vice-president Sebastien Chenu, leaping to conclusions about the nature of the attack, earlier in Parliament warned of “the Islamist threat on our country”.
France has been rocked by a series of jihadist attacks in recent years.
In 2016, a Tunisian man ploughed a 19-tonne truck into crowds in the southern city of Nice, killing 86 people. The Islamic State group claimed responsibility for the attack.
France will next week mark a decade since the attack on the Bataclan concert hall and other locations in Paris, which were also claimed by IS and killed 130 people.
– Agence France-Presse