The supreme Self is neither born nor dies.
He cannot be burned, moved, pierced, cut nor dried.
Beyond all attributes, the supreme Self
Is the eternal witness, ever pure,
Indivisible and uncompounded,
Far beyond the senses and the ego.

The supreme Self, the spiritual Self is immortal.  It cannot be killed, but it can be ignored or corrupted.  There are people who spend a great deal of time and money searching for immortality, but it there all the time, available to everyone.  We just have to develop it.