C172 Flight Instructor reported after initiating a go-around from an unstable approach the aircraft would not climb so they landed the aircraft in a nearby field. No one was injured and only slight damage to the aircraft.

Narrative

On Day 0; I was acting as flight instructor for a commercial 141 student; who is transitioning from an Aircraft Y to a Aircraft X. The aircraft involved in the incident was Aircraft X. All required documentation was on board. At the time of the event; the student had approximately 30 hours in the Aircraft X; and I had more than 70 hours in type and 16 hours in this specific aircraft. After the appropriate pre-flight inspection; we departed ZZZ1 to conduct a familiarization flight to prepare the student for taking his commercial check-ride. Prior to departure I filed VFR flight plans to and from ZZZ1 with the intention of doing pattern work at ZZZ.Upon arriving at ZZZ; we completed 1 full stop landing and 2 successful touch and goes; all without incident. Following those maneuvers; I instructed the student to conduct a power off 180; per the commercial part 141 syllabus; using Runway XX. As prescribed in the POH for the Aircraft X; once we were abeam the numbers and at 1300 ft.; the student applied carburetor heat and then reduced power to idle to simulate an engine out scenario to set up the student's attempt to land within 200 ft. of a predetermined point; which was in this case the 1000 foot markings. Following the simulated power loss; the student immediately maneuvered toward the runway and selected 10 degrees of flaps. The student recognized that his altitude was too high and responded by slipping the aircraft. Note that the POH states that slips may be performed with 10 degrees of flaps. I monitored our position and airspeed and allowed him to maintain the slip until it became apparent that that he would not be able to safely land the aircraft within the prescribed distance from the selected point of touchdown and at a safe landing speed. It was at that point that I called for a go-around; which prompted the student to add full power and configure the aircraft accordingly; including the application of full power and removal of carburetor heat. Despite the student's efforts to commence his climb; the aircraft maintained a nose down attitude. Recognizing that he was unable to pitch the aircraft for climb despite his efforts to do so; the student stated that something was wrong with the aircraft; and that it would not pitch up to climb.I announced that I had the aircraft and took control; applying unusually strong back pressure on the yoke. I immediately noticed that the nose was extremely heavy beyond anything I had experienced in this airplane prior to this flight and the flight characteristics were abnormal. With much effort and using both hands to strongly pull back on the yoke; I instructed the student to push the trim wheel toward the floor; which had little to no effect on the plane. The flaps were not retracted from the 10 degree position as we were not able to establish a positive rate of climb and needed to avoid a loss of lift. At this point; we were nearing the departure end of the runway at 439 ft.; about 120 ft. above field elevation; so I was focusing on a plan that would permit as safe a landing as possible despite the continued problem with the flight controls. Given the problem with the controls; the persistence of the strong nose down tendency; and the low altitude; I determined that an attempt to execute a 180 degree turn back to the runway would have been ill advised and dangerous. I identified a field about 45 degrees to my right and aimed for it; positioning the plane to safely make an off airport landing if full motion of flight controls could not be restored. With the aircraft already exhibiting malfunctioning control surfaces; I decided; rather than turn away from an available safe landing environment and risking a stall-spin; that the best possible outcome would be to put the airplane down in the field.As I was nearing about 100 ft. above the initially chosen field; I realized that it contained a fence could interfere our roll out; so I added power to be able to hold altitude long enoughto go over the fence and toward the next immediate field. We came down; landed uphill; and rolled out to a safe stop. There were no injuries. After rollout; I secured the aircraft by cutting the mixture to shut the engine down. The student and I exited the aircraft to assess any damage sustained during the landing. Initially we found none; until it was noted [we had] a bent nose wheel frame; a common occurrence with soft field landings. Following the initial inspection; I called the front desk of the flight school to inform them where I had safely landed. I also called the owners of the flight school; and then called emergency services.