Some WWII era pilots thought they'd broken the speed of sound in a steep dive but those planes simply had too much drag to do so. The pitot/static system (used to measure airspeed) was also found to be inaccurate at very high speeds, making the airspeed indicator read much higher than was actually the case.

Most propellors lose a great deal of efficiency when the tip speed exceeds about Mach 0.9. Specially designed props like the one on the XF-84H and the Tu-95 can operate higher but the sound levels from the shock waves are incredibly loud, causing severe problems. Even for some light planes whose props exceed about Mach 0.9, the noise levels get very high, making them unpopular in populated areas.

In general, the answer to the OP appears to be no, it isn't possible to use a propellor to get a plane past Mach 1 in level flight.