Banbury cakes are very similar to Eccles cakes but are much older. Eccles cakes are round, while Banbury cakes are oval. A Banbury cake is certainly not a cake or a bun, but rather a sweet pastry.
In The Penguin Companion to Food, I found that local archives show that Banbury cakes were sold in 1638 by one Bette White in 12 Parsons Street in the town of Banbury in Oxfordshire. In the early 19th century, Banbury cakes were baked and sold by two sisters, Lizzie and Lottie Brown. Florence White writes in her Good Things in England from 1932 that the best Banbury cakes can be found at E.W. Brown’s ‘The Original Cake Shop’ on 12 Parsons Street, the exact address where they were sold nearly 300 years earlier. Banbury cakes were sold by street vendors from wicker baskets lined with white cloths. For the town of Banbury, this meant that making these special baskets was an additional cottage industry. Eventually cardboard boxes made into the shape of the cake shop took the place of the baskets, and Banbury cakes were sent all over the country with the postal coach.