No, I don't think they do. You could mount it on a left handed base but the swivel would be to the outside. I think it would work without problems but that's just thinking not trying.
If your Chet Atkins Bigsby arm is stationary and you need it to swing you need to change the bracket. The part you need is P-GB-BRACKET-1X. Its a straight swap. You can get the part at Antique Electronics Supply - https://www.tubesandmore.com/products/bracket-bigsby-chet-atkins-2-pieces
Its US$34 but the only way.