Yes, there is such an idea as spiritual death in God's Word. It is specifically about the "second death" of Rev.20, of being cast into the "lake of fire" at the end of Christ's future "thousand years" reign. And that is prophecy.
John 3
3 Jesus answered and said unto him, "Verily, verily, I say unto thee, Except a man be born again, he cannot see the kingdom of God."
5 Jesus answered, "Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.
6 That which is born of the flesh is flesh; and that which is born of the Spirit is spirit."
This is a very, very simple matter.
Our flesh is born of water (in woman's womb; that which is born of flesh), but our spirit comes from God (Eccl.12:5-7). Our spirit must... be born again (from above) by The Holy Spirit to enter into the Kingdom of God.
That is done by confessing our sins to Christ Jesus, and confessing Him as God's promised Saviour, and believing by Faith that He died on the cross and The Father raised Him from the dead, to offer us remission of our sins past.
In Matt.23:27 when Jesus spoke of the unbelieving corrupt scribes and Pharisees, that they were like "whited sepulchres, which indeed appear beautiful outward, but within are full of dead men's bones, and of all uncleanness", that showed this idea of not being born again by The Spirit, and thus they represented the spiritually dead.
In the time after Christ's future second coming, these type of souls that are spiritually dead will still exist throughout His thousand years reign with His elect. These spiritually dead (not born of The Spirit) will be subject to perishing the "lake of fire" at the end of that thousand years, and that is called the "second death", the death of one's soul and spirit, not one's flesh.