Two of the people hit by the motorist, who was thought to be drunk, were in serious condition after the motorist ran into them on Sunday night.
Police arrested the 40-year old and said the man could face charges of terrorism.
This incident comes only days after a man had been shot and killed after he entered a police station with a knife slashing two police officers while proclaiming his love for Islam.
A police source speaking about the attack on Sunday in Dijon said the man had used a Clio, which is made by Renault, to run the people over before being stopped.
Pictures of the accident showed injured people lying along the roadside while emergency personnel worked on them. Official from France’s anti-terrorism unit arrived to start an investigation into the incident.
Amongst the injured was a child who is 11, according to media in the area.
Police have known of the man and said he might have carried out the attack to copy ones that have been carried out in both France and Australia in recent days.
A spokesperson for the Interior Ministry in France said witnesses heard the man use expressions such as Allahu Aknar or God is Great.
He was heard to say the phrase “For the Children of Palestine,” which was in reference to the deaths of over 500 Palestinian children during the 50-day war between Palestinians and Israel this past summer.
French officials believe the man was acting alone but have not yet interrogated him.
Officials also said the man whose name has not yet been released could face charges of terrorism but the justice system would decide that.
Witnesses believe the driver had been in route to a local police station but only reached the area’s surrounding streets before he had his car stopped.
Security in France has been stepped up after a man converted to Islam who was wielding a knife was shot and killed on Saturday after attacking three police.