The car sped off after the incident described by local mayor Patrick Balkany as “without doubt a deliberate act”.
The Paris prosecutors’ office said its anti-terrorism unit has launched a probe into “attempted killings... in relation to a terrorist undertaking”.
Defence Minister Florence Parly condemned the car ramming as a “cowardly act”, saying it did “nothing to dent soldiers’ determination to work for the security of the French people”.
“Security forces are actively seeking the perpetrator who is on the run. The ongoing probe will determine his motives and the circumstances in which he acted,” Parly said.
France, which has been under a state of emergency since the November 2015 attacks in Paris, has seen a string of assaults on security forces, including those guarding key tourist sites.
Parly said three of the soldiers were “more seriously injured”. All six were hospitalised, but none have life-threatening injuries, Balkany said.
They are part of the 7,000-strong anti-terrorism Sentinelle force set up after the November 2015 bombings and shootings that killed 130 people in the French capital and were claimed by the Islamic State (IS) group.
IS has repeatedly said that France is a target because of its participation in the US-led international coalition fighting the jihadist group, with French jets carrying out air strikes against the extremist group in Syria.
Today’s incident took place at about 8:00 am (1pm Thai time) outside the barracks in the northwestern Paris suburb of Levallois-Perret.
“This morning, as a group of our soldiers were leaving their barracks to go out on patrol, a BMW vehicle parked in the alley rammed into them,” Balkany told news channel BFMTV, adding that the car had “accelerated very fast”.
The incident came just four days after Sentinelle soldiers arrested an 18-year-old with a history of psychological problems at the Eiffel Tower where he brandished a knife and shouting “Allahu Akbar” (God is Greatest).
He told investigators he wanted to kill a soldier, sources close to the case said.
In February, a man armed with a machete attacked four soldiers on patrol at Paris’s Louvre Museum, while in April another extremist shot and killed a policeman on the Champs Elysees, the French capital’s most famous boulevard.
In June, a 40-year-old Algerian doctorate student who had pledged allegiance to IS attacked a policeman with a hammer outside Notre Dame cathedral.
The wave of attacks in France has had a serious impact on tourism in the world’s top tourist destination, but the industry has begun to recover as incidents have become more widespread and generally less deadly.
With terror attacks hitting not just France but also Belgium, Britain and Germany, potential travellers show “a kind of fatalism”, said Josette Sicsic, head of Touriscopie, a firm that tracks tourist behaviour.
The French tourism ministry expects a five to six percent increase in overall arrivals to France this year, for a new record of 89 million visitors.
The lowest point for Paris came at the end of March 2016 – four and a half months after the Paris attacks when IS jihadists targeted people enjoying an evening out at trendy eateries, a concert hall and the national stadium.
In a rebound that began at the end of 2016, Paris saw a record 2.6mn foreign arrivals in the first four months of this year – a 19% increase over the same period last year.