Again, Lewis is covering quickly a theological doctrine that is complex and controversial. Not all Christians would agree that "a Christian can lose the Christ-life which has been put into him." What we're talking about here is the doctrine of the perseverance of the saints, or eternal security.
Those who believe in eternal security believe "once saved, always saved." That is, once someone has truly accepted Christ personally as his Savior, he will go to heaven when he dies; nothing he does after that point (including falling into sin) puts his eternal state at risk. They base this belief on passages such as John 6:37-40, John 10:27-28, 1 John 2:19, and others.
Other Christians would teach that if a person sins willfully and unrepentantly after salvation, he can lose his salvation. (Those who believe in eternal security would counter that a person who calls himself a Christian but lives that kind of life casts doubt on whether he was actually saved in the first place.) People who believe you can lose your salvation base their belief on passages such as 1 Timothy 1:18-20, Hebrews 6:4-6, and Hebrews 10:26-27.
Obviously, Lewis was in the second of these groups. His position on this topic crops up again in The Screwtape Letters, The Great Divorce, and the last book of the Narnia series, The Last Battle (in which we learn that Susan, who had previously served Aslan, no longer believes in Narnia).