Police were scrambling to track down the vehicle, which took off after the incident described by local mayor Patrick Balkany as "without a doubt a deliberate act".
The incident took place at about 8:00 am (0600 GMT) outside a military barracks in the northwestern Paris suburb of Levallois-Perret.
Balkany told the all-news channel BFMTV that the car "accelerated very fast when they (soldiers) were coming out" of the barracks.
France has been under a state of emergency since November 2015 and has seen a string of attacks on security forces, particularly those guarding key tourist sites.
String of bloody attacks
In the bloodiest attack targeting France, 130 people were killed in a wave of shootings and bombings in Paris on November 13, 2015, in carnage claimed by the Islamic State group (IS).
In January 2015, two brothers who had vowed allegiance to Al-Qaeda gunned down 12 people at satirical weekly Charlie Hebdo in Paris.
France is part of the US-led international coalition fighting IS and has carried out air strikes against the extremist group in Syria.
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.
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 attacks have taken a serious toll on tourism to France, 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 in 2017.
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 2.6 million foreign arrivals in the first four months of this year -- a 19 percent increase over the same period last year.