District Councils in Wiltshire
I found it a little difficult to find out the district councils of Wiltshire, please find the below list. Wiltshire Council is a unitary authority. Although Swindon is technically with Wiltshire it is autonomous in terms of governance (I think).
Kennet District Council - (East)
Salisbury District Council - (South)
West Wiltshire District Council - (West)