An IEEE Spectrum article explains that PROP-M was never deployed:
Sadly, PROP-M never got a chance to do any exploring. The Mars-3 lander separated from its spacecraft on December 2, 1971, and entered the Martian atmosphere. After aerobraking with a heat shield, the lander deployed a parachute to slow itself down, and then when within range of the ground fired its retrorockets and made a successful landing on the surface, impacting the ground at about 20 meters per second and cushioning itself with shock-absorbing foam. Four petals on top of the capsule opened and the lander began to transmit data 90 seconds after landing, but all contact was lost just 20 seconds later, before the PROP-M rover had a chance to deploy.
If you just read this portion, there might still be some uncertainty as to whether it could have deployed on its own, but the explanation of how the rover was intended to work clears that up:
The way that PROP-M was intended to work was that after landing, a signal from Earth would instruct the Mars-3 lander to place the rover on the surface with a robotic arm
With this, there can no longer be any uncertainty. The rover did not have an automatic deploy function. It was awaiting a signal from Earth, thus since communication was interrupted it could not have deployed.