As reported to the FAA. Reports are unverified, and codes are left as the FAA publishes them.
What was found
WHILE CLIMBING TO ALTITUDE THE CABIN ALT WAS STILL CLIMBING AT 700 FT/MIN. LEVELED ALT AT 390 & THIS SHOWED NO CHANGE IN CABIN ALT, TRIED TO CHERRY PICK THE CABIN DOWN WHICH ALT HAD NO EFFECT. FL350 WAS THE LOWEST ATC COULD ASSIGN US SO WE STARTED A DESCENT, ONCE AT 350, THE CABIN LIGHT CAME ON (THE CABIN WAS STILL CLIMBING). STARTED EMERGENCY DESCENT. ALL CREW ARRIVED SAFELY BACK TO BASE.