Hadn't heard about Whipcar - interesting idea. But I haven't heard of anything similar here in France.
Paris has just recently announced that they are setting up an "Autolib" (similar to their Velolib arrangement for bike sharing) but it's definitely a city-run venture, with (I think) small electric vehicles. Autolib
OK, anyone heard about private car rentals anywhere in France?
Are you talking about private car hire or car-sharing? If the latter there's a site - covoiturage.fr (I think- you might have to Google it) which links people going somewhere with people wanting to go somewhere, whether on a one-off basis, or regularly like parking somewhere outside town and taking one full car into town for work.
My daughter travelling down to Nice the other day (own car) met up with someone wanting to go to Nice from near here. The site "vets" the registered users, and also calculates what the "passenger(s)" should pay in contribution to tolls, petrol etc in the context of type of car, distance etc. Seems to work reasonably well. (Daughter got 50E which represented about a third of her actual cost, but in theory she could've taken 2 or 3 more people - actually she couldn't 'cos she was going back to Corsica fully laden, but nevertheless was a nice help with her expenses.)