Origin, Meaning of the word 'Bailey'

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.