there are different kinds of immortality or what Buddha termed "the deathless"
there is the immortality of the lustful materialists, such as Wafflehead or the Pharaohs of Egypt, that believe they, personally, will live for ever
then there is the deathless of the Enlightened Ones, that have minds that do not hold to anything as being "theirs". thus, when there is nothing to lose, i.e., nothing to die, this is the deathless
when the mind thinks with angst: "I will die", this is "death"
when the mind thinks: "the mind & matter that compose this life will end" this is the deathless or immortal state of mind