This house is called the Old Guildhall and there are various references to it on the web, mainly because it provides - or once did - bed and breakfast accommodation. It has obviously been heavily modernised. There is no mention in Pevsner, it is not on the list of Gildhalls in PSIAH Vol XiX (see References), nor is there a gild mentioned at all in Badwell Ash in that list (although it is likely there was one). Furthermore it is not near the church. There is nothing in the Listings. The sale details in 2020/21 make no mention of the history. So, no evidence - just the name. A very unlikely candidate for a Gildhall.
Another view with the name visible on the wall to the left of the white plant urn.