What is self?

Praise be to Allah, The Lord (The One and Only) of the Worlds, and blessings and peace be upon our Master Muhammad, the Truthful and the Faithful.

Originally, man is but self; it is his being, and it is the one that is Divinely assigned and tried, and it is the self that is obedient, disobedient, superior, degraded, loving or resentful. The heart is the center of the self, and man is but this self that never dies:

 ﴾Everyone shall taste death﴿ 

[Aal-'Imran, 185]

Testing death is different from death, so the self tests death but never dies.
 

﴾ "O (you) the one in (complete) rest and satisfaction!﴿

[Al-Fajr, 27]

﴾He is the All-Knower of what is in the breasts (of men).﴿ 

[Al-Mulk, 13]

Man is but an everlasting self that is Divinely assigned, it will be brought into account, it will be punished or rewarded and it will be prosperous or miserable. In other words, the self is man him/herself, given the body is the shell and the soul is the driving force.