The Spirit is called eternal in Hebrews 9:14.
"For if the blood of goats and bulls and the ashes of a heifer sprinkling those who have been defiled sanctify for the cleansing of the flesh, how much more will the blood of Christ, who through the eternal Spirit offered Himself without blemish to God, cleanse your conscience from dead works to serve the living God?" (Hebrews 9:13-14)
While the term "eternal" is not used in every verse, the concept that the Holy Spirit is eternal is clearly taught throughout the Bible.
Beginning in Genesis, the Spirit of God participated in creation:
"The earth was formless and void, and darkness was over the surface of the deep, and the Spirit of God was moving over the surface of the waters" (Genesis 1:2).
The Spirit of God makes people:
"The Spirit of God has made me, And the breath of the Almighty gives me life." (Job 33:4)
The Spirit of God is everywhere present:
"Where can I go from Your Spirit?
Or where can I flee from Your presence?
If I ascend to heaven, You are there;
If I make my bed in Sheol, behold, You are there.
If I take the wings of the dawn,
If I dwell in the remotest part of the sea,
Even there Your hand will lead me,
And Your right hand will lay hold of me." (Psalm 139:7-10)
The Spirit gives eternal life. If the Spirit Himself is not eternal, then how can He give eternal life?
"For the one who sows to his own flesh will from the flesh reap corruption, but the one who sows to the Spirit will from the Spirit reap eternal life." (Galatians 6:8)
Jesus said: "I will ask the Father, and He will give you another Helper, that He may be with you forever; that is the Spirit of truth, whom the world cannot receive, because it does not see Him or know Him, but you know Him because He abides with you and will be in you." (John 14:16-17)
If the Spirit is not eternal, how can He be with us forever?
But we have the eternal Holy Spirit dwelling in us when we believe in Jesus.
“And you also were included in Christ when you heard the message of truth, the gospel of your salvation. When you believed, you were marked in Him with a seal, the promised Holy Spirit, who is a deposit guaranteeing our inheritance until the redemption of those who are God’s possession.” (Ephesians 1:13-14)