There were indeed several full-auto variants, notably the M712 Schnellfeuer (rapid-fire) model, as well as full-auto C96 copies made in Spain and China in the late 1920s and throughout the 1930s.
The Chinese were very fond the C96 (Which was known as the "Box Cannon" in Chinese) and I'm told one of the popular tactics for employing the full-auto Broomhandle variants during the Warlord/Chinese Civil War-era was to hold it sideways (not dissimilar to what's known as "Gangsta Style" nowadays) and then pull the trigger, letting the recoil right the gun as it sprayed bullets everywhere.
Bear in mind that the sub-machine gun was a relatively new concept at the time and there were only a couple of practical designs available - the Thompson M1928A1, the MP18 and the Suomi KP-31 were about it until pretty much right before World War II - so in that context, a full-auto Mauser Broomhandle wasn't nearly as silly or impractical as it appears to us today in hindsight.
On an unrelated note, what's with the ",,," thing? It's not a grammatical convention I have encountered before.