CE-560 Captain reported pressurization control issues with an excessive cabin altitude rate of climb with a differential pressure of 8.9 PSI. After descending to a lower altitude; the flight crew returned to the departure airport.

Narrative

First pressurization issues was noticed during our first level; the cabin rate of climb spike between 5;000 and 6;000 FPM. This happened two other times between FL230 and FL330. This issue happened each time we made a power reduction upon reaching our assigned altitude. We did try making adjustments by increasing the rate of climb but there was no change in the cabin altitude. During our climb to higher altitudes we noticed that the cabin altitude was between 4;000 to 5;000 feet but the differential higher than normal. As we climbed to fl330 the cabin altitude still did not increase beyond 5;000 feet but the differential pressure was right at maximum 8.9 PSI. After discussing our options on how to proceed with this issue we initially decided to stay FL330 to assess the situation.Shortly after leveling off at FL330 we decided descend to a lower attitude to try to lower the differential altitude. During the descend the cabin ALT light came at which time we asked for an immediate descent from ATC; performed the appropriate checklists; and requested a return direct to ZZZ. During our descent we concluded that pressurization in the cabin was controlled and that we didn't need any assistance upon landing but did let ATC know that we would be making a landing at ZZZ and would keep them informed if anything was to change. We did inform operations and maintenance control of our situation and our intentions to return to ZZZ.