The Origin and Meaning of the word 'Bailey' derives from the Old French word 'baille' meaning an enclosure

So pretty safe to link this to the introduction of "Norman" words / phrases, and we know that the Norman lords used the Forest as a hunting ground for Deer and Boar.

The Forest is divided into many enclosures , or bounded areas, so it should not be a surprise to see the word Bailey in the area too, both in place names and as a surname.