Yes, they do.
There is close to a total ban on handguns in private hands, but it's not actually absolute. It is close enough that in media shorthand, it is not surprising it is referred to as a ban.
There are a few exemptions for 'genuine' collectors - but you can't just say you are a collector, and there's also provision for a person to have what is referred to as a licence for a 'personal protection weapon', in cases where there are special reasons, normally connected to previous government service, to think a person's life might be in specific danger. I have a PPW permit, but I don't expect to be allowed to renew it, because they keep tightening the regulations, and because so much time has elapsed between the reason I got it and today, that it's unlikely I'll convince the authorities, there's any danger left. The number of PPW licences in England is somewhere in the low thousands from what I have heard - or about 1 in 10,000 people at most. So it's extremely close to a ban, but technically isn't quite there.
Thanks for taking the time to explain the situation.