SR20 Flight Instructor reported engine oil system problems while in flight. Engine problems became severe; forcing the flight crew to make an off-airport landing.

Narrative

During preflight the oil quantity was 6 quarts hot. Oil Temp was ~160 degrees during engine start. Run up was normal. RPM drop was within limits; oil pressure within limits; everything else was also normal. Taxi and takeoff were normal with the student pilot flying. We proceeded on the local crossing departure with autopilot engaged. South of ZZZ1 we started climbing from 2;500 feet to 4;500 feet. We stopped the climb at 2;900 feet to clear the Bravo Shelf; and then resumed climb with full power; with the fuel pump on; mixture top of green. During the climb; the student pointed out the oil pressure indicator was below 50 psi and in the yellow; then CAS annunciated about the oil pressure. We continued to monitor the oil pressure. The oil pressure started dropping again we leveled off at 4;400 feet and immediately started looking for an alternate airport. Oil Pressure was decreasing every second. Initially we pickedZZZ2; however we determined ZZZ was closer and turned direct that way. CFI made a call on the practice area and then switched to ZZZ frequency. Meanwhile; we started descending to 2;500 feet at ~600 feet FPM. It appeared to us that the descent slowed at ~3;100 feet without any input by the crew. At that point the Instructor took controls; with oil pressure in the single digits. The CFI made one call to ZZZ CTAF indicating we were enroute with low oil pressure and would be taking any runway. We continued with the descent with low power setting and the RPM went to 2800+; oil temperature went to 200+; oil pressure ~2 psi and red CAS message annunciation. We talked about CAPS but determined we were too low. At that point around ~600 feet AGL we experienced severe engine roughness and heavy vibrations. The airport was not in sight and we knew we would not make the airport. The student pointed out a field with three dirt roads on it. First road we were lined up for had power lines so we side stepped to a road on the right; which had a vehicle on it. We side stepped again and configured the aircraft for landing. The student called out the airspeeds and the CFI landed on the road at normal approach speed; avoiding any obstacles. The engine quit and propeller stopped spinning on the landing rollout. We stopped the aircraft and the student secured the aircraft with throttle idle; mixture cutoff; fuel pump off and fuel selector to off. Student set the parking brake; we determined the aircraft was secure and we exited. After exiting; we observed oil all over the bottom of the fuselage; also leaking through the engine cowling and within the engine compartment. The CFI checked the oil dipstick and determined the cap was tight.