Angela is right about lemmings in the high sub-Arctic at least, and she is right about their being rodents, but a zoologist might take issue with her expression "like mice". Mice and lemmings are not very closely related. We speak of mice as being murine rodents and lemmings as microtine rodents. They differ quite a lot, say, much as weasels differ from badgers. This is just an analogy you understand, but it should give you some idea.
Now, I had never thought of it so specifically before, but it is interesting that lemmings and arctic ground squirrels are almost the only rodents of the far northern tundra (probably I am overlooking one or two, but feel welcome to correct me).
Certainly one can see all sorts of reasons why mice and rats might not be happy there, but they seem to have adapted everywhere else. Possibly it has something to do with the fact that murine rodents don't generally seem to hibernate.Mice certainly have adapted to deep freezers used for storing food in temperate climates, even developing local strains with heavy builds and thick coats. Perhaps more significantly, they have become a source of serious ecological concern on polar and sub-polar islands, where they have immigrated as stowaways on the ships of whalers and scientists.
There appears to be no equally strong trend on mainland areas, but I have no information about mice on mainland Antarctica, nor on Greenland for example. I would be happy for some pointers.
It is easy to rationalise why mice have survived on, for example, Svalbard and South Georgia islands, given the presence of shelters and food from humans, but I am not sure that that is anything like the whole story. On the islands the problem of immigration might simply be that there was no hope of survival during the ice ages, and no opportunity (before humans obligingly offered lifts) since then.
Ideas anyone?
Jon