Ruger made/makes a MKIII in .22lr...... Browning never did.... Brownings .22lr pistols are called Buckmark.
Either way both are really good guns and if you can get one for around $250 to $350, depending on the shape it is in and the extras ie: extra mags & mag loader, I would not hesitate.
Bear in mind that a new Buckmark is in the $450+ range, and a new MKIII is in the $450+ range as well, Mags are about $30 new and a mag loader is $30 new as well.
Its worth a look by all means.
Cheers!