Now many have thought that the Holy Spirit is the ointment of Christ. And well it is said ointment, because He is called the oil of gladness, the joining together of many graces giving a sweet fragrance.
But God the Almighty Father anointed Him [Christ] the Prince of priests, Who was, not like others anointed in a type under the Law, but was both according to the Law anointed in the body, and in truth was full with the virtue of the Holy Spirit from the Father above the Law.
This is the oil of gladness, of which the prophet says: “God, even Thy God, hath anointed Thee with the oil of gladness above Thy fellows.”
Lastly, Peter says that Jesus was anointed with the Spirit, as you read: “Ye know that word which went through all Judea beginning from Galilee after the baptism which John preached, even Jesus of Nazareth, how God anointed Him with the Holy Spirit.” The Holy Spirit is, then, the oil of gladness.
And well did he say oil of gladness, lest you should think Him a creature; for it is the nature of this sort of oil that it will by no means mingle with moisture of another kind. Gladness, too, does not anoint the body, but brightens the inmost heart, as the prophet said: “Thou hast put gladness in my heart.”
[...] And well is that called oil of gladness wherewith Christ was anointed; for neither was usual nor common oil to be sought for Him, wherewith either wounds are dressed or heat assuaged; since the salvation of the world did not seek alleviation for His wounds, nor the eternal might of His wearied Body demand refreshment.
Nor is it wonderful if He have the oil of gladness, Who made those about to die rejoice, put off sadness from the world, destroyed the odour of sorrowful death. And so the Apostle says: “For we are the good odour of Christ to God;” certainly showing that he is speaking of spiritual things.
But when the Son of God Himself says: “The Spirit of the Lord is upon Me, because He hath anointed Me,” He points out the ointment of the Spirit. Therefore the Spirit is the ointment of Christ.
Or since the Name of Jesus is as ointment poured out, if they wish to understand Christ Himself, and not the Spirit of Christ to be expressed under the name of ointment, certainly when the Apostle Peter says that the Lord Jesus was anointed with the Holy Spirit, it is without doubt plain that the Spirit also is called ointment.
Ambrose of Milan (c. 337-397): On the Holy Spirit, 1, 10, 100-104.