An air carrier First Officer reported during push back the tug and aircraft slid on ice and the tug hit the aircraft.

Narrative

Ground Control advised to push at our discretion. The Captain and the tug driver exchanged the proper callouts to begin the push; including the parking brake being released and the tug driver clearing the start of the engines. As the push began; I started the engine as directed by the Captain. As I was heads down starting the engine; the tug driver stated to the Captain that the tug had applied the brakes; and the tug and the aircraft were sliding. The Captain responded; asking if the tug driver would like the aircraft parking brake [set]. The tug driver responded yes and when the parking brake was set the aircraft stopped moving. The Captain and I did not notice anything unusual about the push. The tug driver and the Captain then coordinated to disconnect the tug and power back to the gate. The Captain tried to power back to the gate; but was unable and then coordinated being towed back to the gate. The Captain and I were unaware of any damage to the aircraft. During pushback; the tug slipped on ice and hit the aircraft. The area beyond the J-line that we pushed into was not clear of ice before commencing the pushback.