The dramatic chase was captured in an amateur video posted on social media.
The driver ran off when the auto came to a stop but was then arrested after being tackled by officers.
Still chased by police, he then crashed through two glass doors at the airport's Terminal 1 to drive onto the runways.
The Lyon prosecutors' office is investigating possible charges of attempted murder against the suspect.
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An amateur video on social media showed the auto being chased by several police vehicles.
A source close to the case said the man suffered from psychiatric problems but did not have a police record, and was not on the country's watchlist for suspected radicalisation.
His identity has not yet been made clear and his motive is not known.
A passenger waited for his flight, in December 2011, during a strike at Lyon-Saint Exupery airport, central France.
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He then made his way to the lobby, where he was met by police officers who gunned him down. -This report was updated at 11:38 a.m. Emergency officials rushed four more victims to a local hospital, where two were later pronounced dead.
Around 4,000-5,000 passengers are estimated to have been affected by the incident. The airline only had to cancel one United Kingdom flight during the incident.
Officers said the vehicle had come onto the runway during a high-speed pursuit by police cars and a helicopter as it drove in the wrong direction along a nearby motorway.
All flights in and out of Lyon Saint-Exupery airport were temporarily suspended.
Local news reports that the chase had begun on a nearby motorway, where the driver of a Mercedes had been going the wrong way and refused to stop. Nobody was reported injured at the scene. "You should be vigilant in public places and follow the advice of local French authorities", they advise.
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