Flight Instructor on training flight reported smoke in the cabin inflight and an immediate landing.

Narrative

Preflight inspection and aircraft runup were normal with no noticeable discrepancies. Takeoff and landing gear operation was normal on departure. The landing gear was not cycled in the flight prior to the occurrence of the fire. Leading up to the event; there were no abnormal indications throughout the flight. After completing some training maneuvers; I instructed the student to perform an emergency descent. We were approximately 7 nautical miles to the NE of ZZZ at 7;500 feet MSL at 135 KIAS. The student pulled the throttle to idle; lowered the gear handle and began a left 45-degree descending turn while inputting approach flaps. The aircraft accelerated to approximately 145 KIAS. At this time; I began to smell something burning; and at the same time; the flaps had come to a stop. I instructed the student to end the maneuver. By this time; smoke started to fill the cabin. I took controls and brought up the flaps while continuing the emergency descent towards ZZZ Airport. I put the flaps up because I believed the flap motor was causing the smoke. The visible smoke stopped after 30 or 45 seconds and dissipated mostly by the time we were on final approach. The entire event from start to finish was less than 4 minutes. The ZZZ1 Airport Tower frequency was already tuned into the radio; so I [advised ATC] and notified them we had a fire and intended to land at ZZZ Airport. The air traffic controller promptly called the ZZZ FBO manager. I tuned ZZZ Airport's CTAF frequency and called for a straight in approach and landing to Runway XX; winds were 110 gusting 22 knots. I preformed a no flap landing and taxied to the FBO ramp.As we entered the ramp at ZZZ; the FBO manager and one other personnel were on the ramp with fire extinguishers waiting for us. We shut down the aircraft and exited the aircraft. Less than 1 minute later; the ZZZ fire department arrived. They inspected the entire aircraft with a thermal device and stated to not hangar the aircraft until the battery was disconnected. After the inspection; they left.