Broughton Castle is Banbury's strongest nearby sight. Despite the name, it is more a fortified manor house than a castle, with a moat, honey-coloured stone, gardens, and interiors tied to the Fiennes family and the English Civil War.
It sits about 2.5 miles west of the town centre, so it is easiest by car, taxi, or bike. Opening times are limited and seasonal; check before going.