A320 First Officer reported an EICAS message warned of a malfunction with the #1 pressurization system during climb out. Later another warning regarding the pressurization system appeared. The pilots descended to a lower altitude and returned to the departure airport.

Narrative

On climb out; received 'CAB PR SYS1 FAULT' Electronic Centralized Aircraft Monitor (ECAM). SYS 2 worked normally. Referenced the FM and continued. At FL340; received 'CAB PR EXCESS ALT' ECAM with associated master warning. We flew the airplane; silenced the alarm and confirmed the [problem]. The Captain directed me to continue to fly the aircraft and we donned our masks and established comms. He got clearance to descend from ATC and I initiated our descent. The ECAM cleared and reappeared at least once during the descent. During the descent; the Captain let the flight attendants (FAs) know what was going on and they mentioned that the emergency lights came on but that the masks did not drop. We leveled at 10;000 ft. and elected to return back to ZZZ1 since the FAs reported that there were no injuries on board. I coordinated the return with ATC while the Captain informed the passengers of what had happened and communicated with Dispatch. On final at ZZZ1; we received a 'CAB PR LO DIFF' ECAM that cleared on its own after a few seconds. The Captain elected to land the aircraft since we had [priority] and we were overweight by about 5k lbs. We landed without incident and returned under our own power to the gate.