I believe that the play is actually the original, and it became a novel later. The backstory of Peter Pan is used as some of the detail in AS Byatt’s amazing novel The Children’s Book.

I read Peter Pan a long time ago, and I forget the details. The theory I formed at the time was that Peter was a normal boy living a normal life, but this was his dream. That’s why he suddenly disappears for a while every now and then: he’s woken up, and the dream continues without him. I’d need to re-read to see whether that reading actually has any real support in the text: my memories are pretty vague at this point. I do distinctly remember the book as strange, definitely not a normal children’s story.