Some schools will allow you to defer for a couple weeks if you contact them and explain. However, if you try and say "can I defer until Jan 15" or some other significant date that sounds suspiciously like another deadline for another school's deposit, they are likely to say no figuring you just don't want to pony up the cash for deposits for med school. I know, and I agree that non-refundable deposits are total crap and an unethical money maker for med schools, but that's how it is.
You have nothing to lose by asking for an extension.