If we are justified by faith, then how could faith be a free gift? Isn’t the blood of Jesus the free gift?

Faith adds to more faith, and through the Spirit we can have faith added to our initial faith. We add to our faith virtue, then we gain knowledge (2 Peter 1) which leads to higher virtues. All of this is a free gift so that we can be partakers in the divine nature sanctified to eternal life.

That I should be the minister of Jesus Christ to the Gentiles, ministering the gospel of God, that the offering up of the Gentiles might be acceptable, being sanctified by the Holy Ghost.
(Romans 15:16)

The apostles said to the Lord, "Increase our faith!" And the Lord said, "If you had faith like a grain of mustard seed, you could say to this mulberry tree, ‘Be uprooted and planted in the sea,’ and it would obey you.
(Luke 17:5-6)
Notice how the apostles said "Increase our faith".
They had faith. Does this mustard seed of faith grow?

For in it the righteousness of God is revealed from faith for faith, as it is written, "The righteous shall live by faith." For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth.
(Romans 1:17-18)
"from faith for faith"
And my speech and my preaching was not with enticing words of man’s wisdom, but in demonstration of the Spirit and of power: That your faith should not stand in the wisdom of men, but in the power of God. Howbeit we speak wisdom among them that are perfect: yet not the wisdom of this world, nor of the princes of this world, that come to nought: But we speak the wisdom of God in a mystery, even the hidden wisdom, which God ordained before the world unto our glory: Which none of the princes of this world knew: for had they known it, they would not have crucified the Lord of glory. But as it is written, Eye hath not seen, nor ear heard, neither have entered into the heart of man, the things which God hath prepared for them that love him. But God hath revealed them unto us by his Spirit: for the Spirit searcheth all things, yea, the deep things of God. For what man knoweth the things of a man, save the spirit of man which is in him? even so the things of God knoweth no man, but the Spirit of God. Now we have received, not the spirit of the world, but the spirit which is of God; that we might know the things that are freely given to us of God. Which things also we speak, not in the words which man’s wisdom teacheth, but which the Holy Ghost teacheth; comparing spiritual things with spiritual. But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned.
(1Corinthians 2:4-14)

For unto us was the gospel preached, as well as unto them: but the word preached did not profit them, not being mixed with faith in them that heard it. For we which have believed do enter into rest, as he said, As I have sworn in my wrath, if they shall enter into my rest: although the works were finished from the foundation of the world.
(Heb 4:2-3)

And be found in him, not having mine own righteousness, which is of the law, but that which is through the faith of Christ, the righteousness which is of God by faith:
(Philippains 3:9)
Catholic – You say I have a man made doctrine than you quote a man’s doctrines to support your belief that we are saved by works. Isn’t that ironic?