Just finished reading another history of D-Day. For the most part, the BAR was the only automatic weapon the Airborne had for the first couple of days. Not counting the M1A1 or Thompson.
M1A1 is just a folding stock version of the M1 carbine and is a semi-automatic. The M2 carbine (after the war) was a select fire weapon. If you are interested in airborne operations of WWII General James Gavin's book On To Berlin
is a good one.