No. No Elf is immortal. Yes it is true that they have everlasting life and are free from falling ill. However, they are still in danger of being killed by arrows or a sword. Also, if you've read the
Silmarillion, Luthien (who is an Elf [obviously
]) died of grief because Beren was slain. I'm sure, however, that there will be more experienced members who would love to go into greater detail with you about this. I, on the other hand, do not have my book in front of me and can't pull out more quotes and proof and facts to help. So, if anybody would like to take over, be my guest.
