The female protester was struck after a driver surrounded by demonstrators panicked and accelerated.
Some 120,000 "yellow vests", so-called after the high-visibility jackets they are required to carry in their cars, have gathered in 2,000 locations.
They accuse President Emmanuel Macron of abandoning "the little people".
Mr Macron has not so far commented on the protests, some of which have seen demonstrators call for him to resign.
But he admitted earlier in the week that he had not "really managed to reconcile the French people with their leaders".
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