I confess some uncertainty about Paul's intended meaning, but here's what I derive from the context.
Old & New Covenants
As Dave already pointed out, Paul is contrasting covenants--that is the focus of nearly the entire chapter. The last mention of spirit prior to the passage in question is in verses 6 & 8:
6 Who also hath made us able ministers of the new testament; not of
the letter, but of the spirit: for the letter killeth, but the spirit
7 But if the ministration of death, written and engraven in stones,
was glorious, so that the children of Israel could not steadfastly
behold the face of Moses for the glory of his countenance; which
glory was to be done away:
8 How shall not the ministration of the spirit be rather glorious?
In colloquial terms, Paul is discussing what we would call today the "letter of the law" and the "spirit of the law." The law given through Moses was very "letter of the law" oriented, spelling out exceedingly numerous details. The new covenant brought by Jesus focused considerably less on the outward show and much more on the inner person, for example
There is nothing from without a man, that entering into him can defile
him: but the things which come out of him, those are they that defile
the man. (Mark 7:15; see also verses 5-14 & 18-23)
Ye have heard that it was said by them of old time, Thou shalt not
commit adultery: But I say unto you, That whosoever looketh on a woman
to lust after her hath committed adultery with her already in his
heart. (Matthew 5:27-28)
It's not enough to check the boxes on outward appearances; one's heart, might, mind, and soul must be in the right place.
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 (a more expanded version of these thoughts here).
Process vs. Purpose
It is important to note what the Old & New Covenants have in common.
the law was our schoolmaster to bring us unto Christ (Galatians 3:24)
Come unto me, all ye that labour and are heavy laden (Matthew 11:28)
The jots & tittles of how to get there may have been updated, but the destination--and the overarching purpose God has for His children--remains exactly the same.
The Resurrected Lord
Paul uses "Lord" repeatedly to refer to Jesus. The first verse of many of Paul's epistles calls this out specifically, but perhaps most notably Paul says:
5 For though there be that are called gods, whether in heaven or in
earth, (as there be gods many, and lords many,)
6 But to us there is but one God, the Father, of whom are all
things, and we in him; and one Lord Jesus Christ
Unless Paul disambiguates otherwise, I think we're pretty safe to assume that when he says "Lord" he means "Jesus Christ".
On the testimony of Luke 24:39 we know that Jesus has a resurrected body of flesh and bones, and in 1 Cor. 15:53-54 Paul tells us that resurrected bodies are incorruptible.
The word rendered "incorruptible" is ἀφθαρσία (from ἄφθαρτος), which connotes: indestructible, imperishable, undecaying, unending existence. On Paul's own testimony the resurrected Lord is embodied.
These are not the majuscules you're looking for
The capitalization of "Spirit" in its first usage in verse 17 (in most versions of 2 Cor. 3) is something I find unhelpful--the translators have added an interpretation to the text, and I am at least partially inclined to believe their interpretation got it wrong.
When Paul wants to refer to the "Spirit of the Lord" he knows how to do so unambiguously--in fact he does so twice in the next verse and a half; he also did so in verse 3 (I acknowledge, though, that Paul is at least occasionally ambiguous and leaves the readers to piece together his meaning by context).
Since the word πνεῦμα (pneuma) carries a variety of meanings (especially in Paul!) it needs to be disambiguated -- we see Paul doing this 3 times in this chapter, and again for different referents in 2 Cor. 2:13 & 7:13. The question is what context is given for the first use of "spirit" in verse 17? Without context the term is somewhat nebulous.
To give Paul the benefit of the doubt, I conclude that his first use of spirit in verse 17 refers not to the "Spirit of the Lord" at all, but to the same usage of the word found in verses 6 & 8 -- that these verses, which are part of the same thought as verse 17, provide the context we are looking for.
Then in the next two uses of the word he clarifies what Spirit he's talking about, because unlike verses 6, 8, and 17a, he's now talking about a specific Spirit.
- Who is "the Lord"? Jesus
- What is meant by "the Spirit"? That which gives life. In this instance, the new testament through Christ which breathes life into the covenants--its ultimate intent being to transform human beings and raise them to life eternal
- What is meant by "the Lord is the/that Spirit"? (see below)
I understand Paul to be saying not:
- Jesus is a Spirit
- Jesus is the Spirit
- Jesus is that Spirit
But rather he's saying it is Jesus that brings life.