Yes, it does. We know that Adam's spirit died that day and that is the first death recorded in scripture....Spiritual death.
Well, actually zero scriptures refer to Adam's spirit being dead or that he suffered a "spiritual death". What we do know is that God said death would happen the day someone ate of that fruit so a death did occur but scripture does not name it or explain further.
We also know Adam did not physically die so it was a type of death where the person remains physically alive. A body cannot remain alive without a living spirit within them, Jas 2:26, therefore we know that Adam's spirit was still alive as his body was still alive.
This means the type of death was a figurative one. I normally use the term "spiritually dead" to convey that but not to also mean the spirit is literally dead.
Adam was simply symbolically dead in his sin/sins as many people are who are physically alive and have a living soul and spirit.