Ah, I get this question all the time.
The answer is an emphatic no.
The rule is: vermin are permitted, but they have to be smaller than your pinkie fingernail (and non-toxic).
In social situations, rodents (mice, rats and voles especially) can be quite off-putting, if only because they eat the tasty creatures that can make for a nice grooming session.
Quite often I’ll get together with General Coco and Major Bonzo, and our respective spouses; once Major Bonzo had a swarm of killer bees hiding in his goatee, and this was a real damper on the evening. (He was a colonel before this incident.)