"If you want to kill someone without people knowing they were the actual target, wait until they stand next to the President" - Bobby Lee Swagger (could also be a line from that old dude in his cabin)
That's not correct. Immortals can be killed, they just don't die of old age or disease (I am basing this on an amalgamation of many immortals). In the greek pantheon, the gods ate ambrosia which made their blood into ichor, and if they lost their ichor, they could die.