Yes mice can climb up into your bed and bite you while sleeping.
When I was a kid years ago I was sleeping and I felt something bite me and when I woke up I saw a mouse running away from me.
So yes mice can and do sometimes bite people in their sleep so it's important to do all you can do to get rid of mice inside your house to prevent getting bitten.
If a mouse does bite you while sleeping or at any other time then you can get rat bite fever which can be deadly so if you've been bitten by a mouse or mice then you should go to the doctor to get checked out.