When I was a little girl (in the early 80s) I watched an animated film about a fawn who loses his mother during a snowstorm. This is what I remember. They are walking together in the snowstorm looking for shelter. The fawn tells his mom that he's too tired and can't go on. His mother realizes that the cold will kill them but she tells him that it's okay and that he should huddle by her. The storm passes and the little deer wakes up (his mother is curled around him) and he tries to wake her only to find out that she has died (from the cold.)
I don't remember more than that.

Your description prompted me on a short search. I found it!

Nestor, the long-eared christmas donkey. Check it out on wikipedia

That's it! I don't know why I thought it was a deer but when I saw it, I was only 5 or 6. Thank you again!