Cars were effectively banned when cv struck weren't they? (Haven't got a car so don't really know, and I've not followed the thread for a while so don't know whether this has been addressed).
So it is possible - banning them would just extend this.
Use was certainly reduced quite a bit initially. But in fact what happened was that existing inequalities were exaggerated: car owners, after a while, were basically told they could go where they want, while the rest of us were told not to go anywhere. So, car owners could drive hundreds of miles to beaches and the like, and those who didn't own cars could go to a park within walking distance of their doorstep. Dominic Cummings took full advantage of this.
Reasons for discouraging public transport use included keeping trains and buses clear for essential workers to use, protecting staff who work on it, and of course avoiding crowding situations where infection could spread.
All these reasons are fair enough, to some extent. But it was notable that the same thinking was barely applied to car travel. No thought given to the benefits of keeping roads quiet for those whose only choice was to navigate them on foot or bicycle. Nor the impact on all the workers who are needed to keep the road infrastructure functioning. Nor the fact that unlimited travel distances were going to inevitably lead to crowding in certain popular locations.
If I were Prime Minister, I would do the following now, to redress this inequality. I would designate 2 out of 3 weekends for the remainder of the summer as "car free". The rules would be reversed: no non-essential travel by car, anywhere. Those who did not own a car would be given priority access to a system where seats could be booked on trains and buses to anywhere in the UK. There would be a limited number of seats allocated per vehicle, to guard against overcrowding on board. Those who, in addition to not owning a car, live in flats with no outdoors space, would be given free tickets. Britain's non car-owning people could then go and enjoy beaches and natural beauty spots anywhere they liked, and enjoy a taste of what car owners could do throughout the lockdown period. As a bonus, none of these places would have SUVs parked everywhere, and the air quality and noise levels would be even better than usual.