Numbers 30 and 32 The Bank, this is a grade I listed building. I am not surprised.
Dating from the 16th century, with rework every hundred years since. There is a huge and detailed description in tedious prose in the Historic England listing at Link.
The face of the building includes four badly wethered statues of musicians, at least two of which appear to have drone-less bagpipes similar to Cumbrian pipes.
The left hand side includes a magnificent 3 story window bay in ashlar, dating from the 17th century.
On this site www.barnardcastlelife.co.uk it's called the Market Cross. Very interesting images and text. I thoroughly enjoyed looking around. Cheers, Dave