Did you say book? They're printed on paper and don't usually last very well when wet. Some of my books were water damaged. A copy of the Bach Italian Concerto in F got partly soaked in the rain.
I'd play the pieces I remember by ear as much as possible and get into improvisations. There is no guarantee the piano on the island is in good condition with few or no broken key(s) and has been tuned recently. I came acoss old pianos that people gave away that hasn't been tuned for a few years. Even a piano that was played before it was given away the owner didn't spend money for a technician to tune the instrument.
The 2 books that I'd take to the island are the 2 that has many pieces I can play for fun without hours of practice:
- The first is Reader's Digest "100 Favorite Songs of Faith, Friendship & Love". These are old songs from before the 1970s. The pieces have simple LH accompaniment and chords on top that can be read as lead sheets.
- "The Big Book of Classical Music: 100 of the World's Favorite Pieces". These are Classical pieces. Not all were originally for piano such as "In the Hall of the Mountain King" from Peer Gynt Suite by Grieg somewhere in the book.
I like both of these equally well.