'Richard Gillett of Caudle Haddon, Dorset, husbandman, will dated 1 July 1600, probated 16 July 1600 (PCC 55 Wallop): wife Alice; ...'
The will names sons William, Edward "at end of his apprenticeship", and John (executor with Alice); daughter Mary "at marriage"; children of son Richard; son-in-law William Sheppard and his son William.
'This will would satisfy the chronological and geographical requirements for the will of the father of the Rev. William Gyllett, and this testator had sons named William and Richard. The latter, however, was not a beneficiary of the will and is mentioned only as father of grandchildren who were. The presumption is reasonably strong that this Richard was dead in 1600, but his children receive not the £10 which William and Mary receive but only a twentieth as much, that to be divided among them, so it seems possible that Richard was still living, had already married, at which time he had received his portion, though this is not stated in so many words.'
Note that if William was the child of Richard and Alice Gillett of Caundle Haddon, Dorset, he was unlikely to be the child baptized in North Petherton, some 42 miles from Caundle Haddon.