I shared some thoughts in another post about the context preceding this passage--Paul is talking about the spirit of the law. Spirit (pneuma) is that which gives life, and it is the new testament through Christ that gives life, progress, and purpose to the covenants, old and new; He gives life to the ordinances, He gives life to the plan & the people it transforms.
I believe "spirit" in its first usage in verse 17 is used the same way Paul was using "spirit" a few verses earlier. When Paul wants to use the term spirit in a different way (referring to what we would call the Holy Spirit or the Holy Ghost), he disambiguates it from the previous discussion, in both verse 17 & 18 he refers to the "Spirit of the Lord".
The choice to capitalize the first instance of "Spirit" in verse 17 is something I find unhelpful (there would have been no such distinction between majuscule & miniscule in the original manuscript).
From multiple New Testament passages we have clear testimony that Jesus rose bodily from the dead. The passage in Luke is unambiguous in declaring that the resurrected Lord has a physical body (see a more extended discussion in my thoughts on this post).
Paul has already told the Corinthians that resurrected bodies are incorruptible:
53 For this corruptible must put on incorruption, and this mortal must
put on immortality.
54 So when this corruptible shall have put on incorruption, and this
mortal shall have put on immortality, then shall be brought to pass
the saying that is written, Death is swallowed up in victory.
The word rendered "incorruptible" is ἀφθαρσία (from ἄφθαρτος), which connotes: indestructible, imperishable, undecaying, unending existence (see here & here). If Jesus’ resurrected body later decayed, were discarded, or ceased to exist, it wouldn’t be incorruptible.
To be vs. To have
This doesn't mean Jesus does not have a spirit; it means he is not only a spirit. On Paul's own testimony the resurrected Lord is embodied.
As Nihile Sine Deo pointed out here:
[In Luke 24:39] Jesus was emphasizing that He had a body! What He was
NOT saying, is that the body had no spirit in it. Because He clearly
was present in that body. A body without a spirit by definition is a
dead body, a lifeless body, a spiritless body.
Jesus as the spirit of the law
I suggest Paul's meaning is that Jesus brought the spirit of the law, He embodies the spirit of the law, and He is the ultimate manifestation of the spirit of the law. If we want to understand the spirit of the law, look at Jesus. And what is spirit? That which gives life. Jesus gives life to the law, the ordinances, and the plan (a more extended discussion here)
A useful hermeneutic principle is to use clear passages to interpret unclear passages, rather than the other way around. That Jesus rose bodily from the dead--and that his resurrection was not a temporary arrangement--is clearly, repeatedly, and sincerely reported by numerous eyewitnesses, throughout the New Testament.
Although Paul's use of the term "spirit" in multiple ways may be unclear, it certainly does not suggest discarding the testimonies of Matthew, Peter, John, Thomas, James, Paul himself, and many others that Jesus was resurrected with a physical body.
Bodily resurrection is an absolutely central piece of the Christian message--Christianity does not work without the empty tomb. This place of monumental importance is emphasized by the fact that the only detail of the resurrection narrative found in all four Gospels—with no divergence whatsoever—is the fact that the tomb was empty.
Jesus rose bodily from the dead, He took back His body from death, and the victory was permanent. Yes, Jesus has a body.