You must have a definition of "immortality" that is not found in the English dictionary. I won't quote it so as not to raise any straw men. :) In any event, your middle sentence above is internally inconsistent. If you admit that a wicked man "will remain separated from God for all eternity", then you admit that the soul goes on forever and is not annihilated. That is an immortal soul. (And as Harley noted, not eternal, but immortal once created.) You may call this soul "dead to God" if you wish, but that does not affect the fact that such a soul still exists, which is what you deny if you say that the soul is not immortal.
Man's soul was created immortal because it is life which is from God. Had Adam not sinned, he would have never died. Adam's daaged soul is passed on to all of us. God doe snot created damaged souls.
Our source of life is God. Without God we perish. Because we inherit a damaged soul, we are destined to death. God came and made it possible for us to be restored to our original state. In order to do that we must repent of our fallen will, come to Him, and follow His steps.
The unrepentant soul will perish, as Ezekiel states. If the unrepentant departed are to "exist" in a state of complete darkness (separation from Light, which is God), not knowing where they are for all eternity, and you want to call it "life" then, sure, they will be "immortal."