Evidence tending towards the theory that it's a homing mouse: we only took it across the road and didn't drive it some distance away; this mouse also treats the trap as a snack bar; we were mouse-free for a couple of days and now we're not.
So I've done some Googling and found this helpful page of advice.
If it weren't for the fact that we're going away for a weekend in a couple of weeks time, I'd be up to Freshfields to choose our cats pronto.[*] That certainly worked for us back when we got our first cat. Catlet never caught one inside, but she certainly deterred them from venturing out into our living space. I will have a go at bunging up the holes in the airing cupboard at the weekend because I think that must be where it's coming in. Also will try the peppermint oil idea.
[*] Please be assured, we're not just getting the cats as mouse deterrents. This is just the final bit of weight that has tipped the balance in favour of "get cats now" rather than "it would be nice to have another pair of cats sometime."