Man is made up of three parts: body, soul, and spirit. The spirit cannot die, only the soul and body can die. The death you are referencing is the death of the soul. When we “die to ourselves” we surrender our right to direct our own lives and become subject to the rule of the Lord.
Upon bodily death the body returns to dust and the spirit returns to God, assuming the spirit is righteous of course. (Ecclesiastes 12:7).
If that's true then the it is inferred that unrighteous spirits do not return to God. The spirits that do not return to God are sent to hell. In hell, there's no such thing as eternal life for the body, soul, or spirit.
The below verses refer to spiritual death:
Romans 6:23
23For the wages of sin
is death; but the gift of God
is eternal life through Jesus Christ our Lord.
Ephesians 2:1
1And you
hath he quickened, who were dead in trespasses and sins;
2 Thessalonians 1:9
9Who shall be punished with
everlasting destruction from the presence of the Lord, and from the glory of his power;
The only way to avoid a spiritual death is through a spiritual rebirth:
John 3:3
3Jesus answered and said unto him, Verily, verily, I say unto thee, Except a man be
born again, he cannot see the kingdom of God.
1 Peter 1:3-5
3Blessed
be the God and Father of our Lord Jesus Christ, which according to his abundant mercy hath
begotten us again unto a lively hope by the resurrection of Jesus Christ from the dead,
4To an inheritance incorruptible, and undefiled, and that fadeth not away, reserved in heaven for you,
5Who are kept by the power of God through faith unto salvation ready to be revealed in the last time.