Tertullian is not a Church Father and more than Martin Luther is.
The mortality of the soul is scriptural: Ezekiel 18:4 "the soul that sinneth, it shall die." Those who will remain separated from God for all eternity, due to unrepentant sins, hall be dead. Sin is death.
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.
Oh, now we're into quantifying and qualifying who and who is not a church father based upon our own personal preferences??? So, if I understand this correctly, 1) pick and choose which bible verses to believe, 2) pick and choose what theology to believe, 3) pick and choose what church fathers to believe.
No wonder so many are into "free choice".
The mortality of the soul is scriptural: Ezekiel 18:4 "the soul that sinneth, it shall die."
Of course the soul that sinneth, it shall die. Hell is eternal death.
Mar 9:43 And if your hand offends you, cut it off. It is better for you to enter into life maimed than to have two hands to go into hell, into the fire that never shall be quenched