Can propeller driven aircraft sustain speeds at or above Mach 1?
Supersonic level flight?  No.

The fastest propeller-driven aircraft is either the turboprop-powered TU-114 with a measured top speed of Mach 0.73 or the turboprop-powered XF-84H "Thunderscreech" with a design top speed of Mach 0.9 and an actual top speed of either Mach 0.83 or Mach 0.7 (sources disagree).  The latter is an indication of why supersonic propeller-driven aircraft are unlikely: the continuous sonic boom coming off the propeller tips made the Thunderscreech the loudest airplane in the world.

The record for piston-engine aircraft is slightly lower, at Mach 0.71.