A Self in the EBT is by definition that which is permanent/ unchanging, free of Dukkha and completely in/under control, hence whatever is impermanent and unstable cannot be considered as Self (as per EBT definitions).
MN22
But if itâs impermanent, suffering, and perishable, is it fit to be regarded thus:
âThis is mine, I am this, this is my selfâ?â
âNo, sir.â
What you are referring to might, IMO be better considered as the conditioned phenomenon of âPersonalityâ.
It is impermanent, unstable and often the source of much dukkha, when it doesnât live up to the ideals we have set it. 
SA273
Monks, just as two hands coming together produce sound, so, conditioned by eye and visible forms arises eye consciousness, and these three things together are contact. From contact arise feeling, perception, and volition.
âAll these phenomena are not-self, impermanent; they are without a permanent self, not eternal, not stable, changing. Why is this so?
âMonks, these have the nature of birth, ageing, death, ceasing, and rebirth. Monks, all compounded things are as an illusion, a flame, ceasing in an instant; being not real they come (arise) and go (cease).
âTherefore, monks, with regard to all empty compounded things you should know, rejoice in, and be mindful of (awake to) this:
âAll empty compounded things are empty of any permanent, eternal, lasting, unchanging nature; they are empty of self and of belonging to selfâ.