It may well be that C.S. Lewis' quote from Mere Christianity was originally inspired by George MacDonald's words in Sir Gibbie. In his masterpiece of apologetics, Lewis writes,
"Your real, new self (which is Christ's and also yours, and yours just because it is His) will not come as long as you are looking for it. It will come when you are looking for Him. Does that sound strange? The same principle holds, you know, for more everyday matters. Even in social life, you will never make a good impression on other people until you stop thinking about what sort of impression you are making. Even in literature and art, no man who bothers about originality will ever be original whereas if you simply try to tell the truth (without caring two pence how often it has been told before) you will, nine times out of ten, become original without ever having noticed it. The principle runs through all life from top to bottom, Give up yourself, and you will find your real self. Lose your life and you will save it. Submit to death, death of your ambitions and favorite wishes every day and death of your whole body in the end submit with every fiber of your being, and you will find eternal life. Keep back nothing. Nothing that you have not given away will be really yours. Nothing in you that has not died will ever be raised from the dead. Look for yourself, and you will find in the long run only hatred, loneliness, despair, rage, ruin, and decay. But look for Christ and you will find Him, and with Him everything else thrown in."
So it is in finding Christ that we find ourselves:
This means that anyone who belongs to Christ has become a new person. The old life is gone; a new life has begun!
2 Corinthians 5:17.
For you died, and your life is now hidden with Christ in God. When Christ, who is your life, appears, then you also will appear with him in glory.
Colossians 3: 3-4.
For God wanted them to know that the riches and glory of Christ are for you Gentiles, too. And this is the secret: Christ lives in you. This gives you assurance of sharing his glory.
Christians have their identity in Christ. We are God's sons and daughters, as God's Son lives in us.
Yet to all who did receive him, to those who believed in his name, he gave the right to become children of God...
See how great a love the Father has bestowed on us, that we would be called children of God; and such we are. For this reason the world does not know us, because it did not know Him. Beloved, now we are children of God, and it has not appeared as yet what we will be. We know that when He appears, we will be like Him, because we will see Him just as He is.
1 John 3:1-2
If you cling to your life, you will lose it; but if you give up your life for me, you will find it.