Two of the people injured in the car attack in the city of Dijon were in a serious condition, a police source said, adding that the driver had been arrested.
"The man, born in 1974, is apparently unbalanced and had been in a psychiatric hospital," a source close to the investigation told AFP, adding that "for now his motives are still unclear".
The man had targeted groups of passersby at five different locations in the city on Sunday evening in a rampage that lasted around half an hour, the police source said.
"Nine people were lightly injured and two others seriously but their lives do not appear to be in danger," the source added.
Witnesses told police that the driver shouted "Allahu Akbar" and "that he was acting for the children of Palestine", a source close to the investigation said.
Police sources said the driver was known to police for petty offences dating back to the 1990s.
Prime Minister Manuel Valls took to Twitter to express "solidarity" with those injured in the attack.
France is still reeling from a suspected radical Islamic attack on Saturday that saw a French convert to Islam shot dead after attacking police officers with a knife while also reportedly crying "Allahu Akbar" in the central town of Joue-les-Tours.
The assailant, Burundi-born French national, Bertrand Nzohabonayo seriously injured two officers -- slashing one in the face -- and hurting another.
The 20-year-old attacker also cried "Allahu Akbar" during the assault, said a source close to the case speaking on condition of anonymity.
The assault prompted the government to step up security at police and fire stations nationwide.
Nzohabonayo had previously committed petty offences but was not on a domestic intelligence watch-list although his brother is known for his radical views and once pondered going to Syria, the source said.
The anti-terror branch of the Paris prosecutor's office has opened a probe into that attack.
The weekend incidents in France come as governments around the world brace for so-called "lone wolf" attacks by individuals following Islamic State group calls for violence in the countries involved in a coalition fighting the militants in Iraq and Syria.
The group has repeatedly singled out France for such attacks, most recently in a video posted on jihadist sites this week.
Last week in Australia, an Iranian-born with a history of extremism and violence entered a cafe and held people hostage for 16 hours before being killed. Two of the hostages also died.
Last year in France, a recent convert to Islam also stabbed a soldier in the busy Paris commercial complex and transport hub of La Defense.
And the main suspect in the murders of four people at Brussels' Jewish Museum in May, Mehdi Nemmouche, spent more than a year fighting with extremists in Syria.
Authorities in France believe around 1,200 nationals or residents are involved in one way or another in militant networks in Iraq and Syria.
COMMENTS (9)
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There is something seriously wrong will Muslims all over the world. It is time we do some serious thinking into what is wrong with us.
Why do these violent characters move out of their nation? Send them all to Saudi Arabia. Maybe they'll be happy there.
This is getting beyond crazy !!
Allah must be cringing at their cries of AllahoAkbar. For very deluded a person gets as soon as he turns Muslim... something really wrong at the core of our practiced Islam.
How can any rational resident of Pakistan point fingers at Europe, Canada, US and Australia and call them less-tolerant?
All of these countries and regions are significantly more tolerant of minorities than Pakistan. They are not perfect, but they offer much better life for minorities than Pakistan. And what's the worst is that Pakistan does not seem to be improving anytime soon.
Stop looking at other's faults and have a look within your own country and religion.
No capital punishment in France. Alas, make sure he never breathes as a free man again.
@khanzada:
Yeah...that's a good reason to kill people...right?!
@Stranger: Yea....maybe thats why they tried to impose ban on head scarfs
France is a very tolerent country . Please dont test its patience .