The 28-year-old driver, who was arrested immediately afterwards in the city's Blagnac suburb, acted "deliberately" but was not on a list of known extremists, one source said, asking not to be named.
He had previously suffered severe psychiatric problems and was released from a court-ordered hospital stay in December 2016, the Toulouse prosecutor said.
The man had about 10 previous convictions, the prosecutor told the BFMTV news channel.
The investigation is to be handled by regular judicial police rather than Paris's anti-terror unit, he said.
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All three of Friday's victims, who were on a crosswalk when the ramming occurred, were in their twenties, with a 23- year-old woman the most seriously injured and two men aged 22 and 23 also hurt.
The woman's life is not in danger, police said.
France has been victim of a series of vehicle attacks by extremists inspired by the Islamic State group, but there was no immediate confirmation of the driver's identity or motive.
In July last year, a Tunisian man ploughed a 19-tonne truck into a crowd in the southern city of Nice, killing 86 people.
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