Yes and no… Probably not a footswitch, but according to the manual, the NAB version does not have gapless punch in/out, but the DIN model sold in Europe and elsewhere does.
My guess you would use the remote control for this function.
The BR-20 is one model that seems to have all the bells and whistles for every use you can imagine, and yet there are models with little things like this that are incompatible. The manual below might help, but its not perfectly clear either.