Three beginnings
Peter Pan arrives in print in 1902, in The Little White Bird — a novel J. M. Barrie wrote for adults, in which a few chapters describe a baby who flies out of his nursery window and lives in Kensington Gardens.
The play came in 1904: Peter Pan, or The Boy Who Wouldn't Grow Up. It was a sensation, and Barrie kept changing it for years afterwards, which is one reason "the original" is a slippery phrase here.
The novel most people mean is Peter and Wendy, published in 1911, seven years after the play it was based on. So the book came from the play rather than the other way round, which is the reverse of what you would expect.
The hospital
In 1929 Barrie gave the rights in Peter Pan to Great Ormond Street Hospital for Children in London. He asked that the amount it earned never be revealed, and the hospital has honoured that ever since.
When British copyright in the work would otherwise have run out, Parliament added a provision to the 1988 Copyright Act giving the hospital a continuing right to royalties on the work in the United Kingdom. It is an unusual arrangement, made for an unusual bequest, and it means Peter Pan's legal situation genuinely differs from most other classics of its age.
If you have ever wondered why Peter Pan appears less often than Alice or Oz in collections of retellings, that is a large part of the answer.
A stranger book than its reputation
Barrie's Peter is not the cheerful figure of the posters.
He forgets people. He forgets Tinker Bell entirely. He comes back for Wendy after a year, then misses a year without noticing, and at the end of the book he does not recognize her at all — she has grown up, which was never a possibility for him. The narrator calls children "gay and innocent and heartless," and means the last word.
It is a book about the cost of not growing up, written by a man who had thought about that very hard. Read as a children's adventure it is tremendous fun. Read as an adult, at the end, it is quietly devastating.