Evil aliens have decided to kill everyone on Earth! Fortunately, they've given you two minutes to save as many as you can. All you have to do is name a country to save everyone in it.
Among these, at least Latin, Norse, Anglo-Saxon, Brittonic, Slavic, Old East Slavic, Old Ruthenian and Pomeranian (possibly more) didn't actually go extint, just their modern descendants got different names. That doesn't count as extinction in linguistics as far as I'm aware.
I think this list has a pretty infinite number of contenders. middle english, old [insert european language here], proto-indo-european, vulgar latin, celtiberian, a dozen versions of greek. Polari, arguably