Gulfstream IV Captain reported a 'Cabin Press Manual' CAS message accompanied with an aural alarm on climb out. Cabin pressure remained uncontrollable so the crew elected to perform an air turn back and land at the departure airport.

Narrative

I was the PIC (Pilot in Command) on Aircraft X; departing ZZZ; destined for ZZZ1; which departed at XA:30z. Prior to the flight; the crew had adequate rest; though encountered numerous delays and two changes to our destination which let to us landing in ZZZ. Once on the ground we had planned on repositioning the aircraft to ZZZ1. I had gone inside to take care of the bill; while the SIC (Second in Command) had prepared for the flight by conducting the pre-flight and obtaining the clearance.Once I closed the door and secured myself in my seat; I asked the SIC for a brief on what the clearance was. He had informed me of the departure procedure; and that Clearance Delivery also stated that we were to maintain 230 kts. until advised. I looked over the departure procedure and saw no note as to the speed restriction; and the SIC informed me that it was issued with the clearance. I noted that he also put 23;000 ft. into the altitude box on the MCP panel and asked if that was also our top altitude per the clearance.We then started engines and taxied out to Runway XXR. A takeoff clearance was obtained; and we took off. Around 3;000 ft. AGL; we received a 'Cabin Press Manual' CAS Caution Message accompanied with an aural alarm. I maintained positive control of the aircraft and directed the SIC to check the pressurization. He could not control the pressurization; and I then instructed him to contact departure passing 14;000 ft. MSL. As the SIC contacted departure; control informed us that our clearance altitude was 10;000 ft.; and to turn direct to a point on the DP. The SIC acknowledged the clearance and then informed Departure of our situation with the pressurization.The decision to return to ZZZ was made. No other events occurred. As a result; we never received a phone number to contact; nor informed of any pilot deviations. The combination of the day; and stresses of the situation had been contributing factors to the altitude bust; though more importantly; as a safeguard against misunderstanding the clearance; both of us should have been listening together to verify the altitude.