What is the main conflict or problem in the book Peter Pan?
Peter Pan does not grow up, and this creates a poignant dilemma, for everyone around him does. Eventually, if Wendy is any example, the aging grown-ups and Peter become increasingly unable to understand each other. Peter, however, does not want to grow up. He embraces living in an endless boyhood of adventure, which necessitates always finding new, young companions. Further, his clinging to eternal childhood puts him at odds with, and makes him an outlier to, the rest of society.
Yet Peter's enormously vivid imagination allows him to will things into existence. He can fly, and he leads the Darling children on an exciting adventure. A thematic conflict is that of the tension between the child's imagination, which lights up the world, fills it with romance, and makes all things seem possible, and an adult's more measured and less creative but more responsible walk through life.
The main plot conflict arises when Captain Hook and his pirates kidnap the Darlings in revenge for Peter cutting off Hook's arm and feeding it to the crocodiles. This draws Peter into battle. The two adversaries, Peter and Captain Hook, finally square off.