Flight Instructor reported the cabin door separated from the aircraft in flight after the latch became detached. The flight crew returned departure airport and made a precautionary landing.

Narrative

Shortly after takeoff; a 'Door Open;' warning light appeared on the aircraft. As I was sitting in the right seat; I look back to discover the latch partially removed from the door and the door began bouncing and vibrating on its' hinge. I reached back in an attempt to secure the handle into the close position; at that time the door handle became full open and the door began violently bouncing on the safety clip. As I began to hold the handle; the clip sheared off (as there was LLWS/Light Turbulence). At that point in time; the student asked to return to the airport. After some delay in student action; the door became unbearable to hold close and my hand slid off. This caused the door to finally break loose and shear off in flight on the emergency release position. I advised the owner and the door appears to have landed into a lake with no damage or adverse impacts to the aircraft or property damage. After some self assessment; I believe the primary issue resulted in the latch breaking loose from light turbulence. When attempting to close the door and the loosing mechanism happened; after I held the door and saw the emergency clip failing to engage; I knew the door would be lost at that time. No action could be completed other than returning the aircraft safety to the ground.