And this fact, I think, is directly related to the whole issue of the Buddhist "no-self" concept. It is because we die as individuals that we fear death -- the obliteration of the (individual) self is what one's self (using the word in the sense it is used in ordinary language) is constantly trying to stave off, by doing the biological stuff of eating, breathing, fighting off attackers, engendering offspring, etc.
Once one gets the insight that this self which supposes that it is a constantly existing, permanent thing (that is, it would be permanent if it succeeded in winning the fight to keep existing forever, which is what it hopes it can do) is actually merely a set of "heaps" of momentary events, physical and mental, enlightenment begins to dawn.
(At least, that's the theory, on an elementary level.)
^^^^^^^
CB: If no-self means individual members of the species are not immortal, I agree with that.
An interesting side aspect of this is that, mortality of the individuals makes reproduction an important process for the species. If individuals were immortal, having children would be a lesser important matter.
Also, to the extent that people have a desire to live forever, perhaps it may be part of the root of the concept of "living" through future generations, and a way in which the essentially human practice of transgenerational communication through culture was founded. Culture is not only a present generation looking back for ways of life, but a present generation looking to transmit to future generations, as a way to cheat death, perhaps. In other words, the desire to transcend death may have played a role in the origin of humanity for which cultural, socio-historical legacy, is fundamental. Ancestor "worship" and leaving a legacy were probably complementary sides of this fundamentally human characteristic.