neil it is possible, any pest can make bees abscond. not in the dead of winter.
the bees will sting it to death if they can and try to propolize it if they can while not clustered.
mice do make a mess; they chew comb, and frames, consume honey and eat bees and are not very sanitary creatures.
the best defense is to use entrance reducers that are mouse proof or use hardware cloth over the wood reducers to prevent them from entering or chewing their way in.
i have been fortunate not to have suffered mice damage over the years........can't say that about my 4 wheel horse........cha ching.......
what a mess and what an expense! i have seen plenty of others hives who didn't take the precautions to keep them out.
also, if you think you might have a resident mouse and can't open the hive and get to the bottom box (that's where they nest) if it's too cold, i have taken a slim jim and poked it in the hive in a sweeping motion to the back and sides and run it in several times when i saw evidence deer mice were checking out a new home going into winter before i placed the hardware cloth on. they run out rather quickly.
be sure to wear a veil as well..........