You can ride anything on snow and ice if you've a mind to. One winter's eve, they got caught out around here and didn't grit the roads. It was minus 5 overnight, thick ice everywhere, and I had an early train to catch - and a journey to ride at the other end and no chance of taking the mountain bike on the train.
I rode my Brompton to town. It took about twice as long as normal, and on one particularly bad and steep stretch, I thought "that's interesting, no grip at the back. Oh, now there's no grip at the front either. Right foot down, no grip. Left, no grip. Oops this is beginning to feel a bit sketchy". I slowly laid the bike down, and gently crabbed on hands & knees over to the side of the road pushing the bike sideways into the ice so the pedal acted like a crampon.