B767 First Officer reported loss of cabin pressure control in flight. The Flight Crew began an immediate descent and requested priority handling. Cabin altitude remained uncontrollably high so the Flight Crew diverted to make a precautionary landing.

Narrative

This flight consisted of an in-flight [situation] resulting in diversion to ZZZZ. Abeam the northeast coast of ZZZZ1 we gradually lost cabin pressure (with the associated EICAS and visual/audible warnings) and ultimately followed the priority checklist for loss of cabin pressure. This included us both donning oxygen masks and establishing crew communications (in accordance with the priority checklists etc). At the same time; cabin pressure indicated 11;000 feet and slowly rising (less than 500 feet/minute.) In accordance with the checklist we began a descent to 10;000 feet after receiving clearance from ATC. Simultaneously; we made contact with the company over SATCOM for further guidance. I (LC) was the pilot flying and the Captain was PM for the flight. At the time both PACKs were operating normally and seemed to have normal duct pressures; with the cabin outflow valve indicated fully closed. The Captain (to the best of my recollection) switched manually to the other cabin pressure controller and it was around this time that the status message 'cabin alt auto 2' came on shortly followed by 'cabin alt auto 1'. Notably; for the idle thrust descent from FL 380 the cabin altitude rate of climb *increased* significantly (with the reduction in thrust) from that at level cruise flight and the cabin altitude indication approached 14/15;000 feet. In coordination with company we decided that ZZZZ was our best divert field. We landed there without further incident. From the logbook maintenance entry: During cruise flight at 38;000 feet; the Master Warning horn/lights and EICAS message Cabin Alt was on. Cabin Altitude was 11;000 feet. We switched from controller 2 to 1. Had a status message 'cabin alt auto 2/cabin alt auto 1. We descended to 10;000 feet.