Air carrier First Officer reported the nose and right main gear lights did not extinguish upon raising the gear after takeoff; and subsequently also received a Hydraulic System 3 failure alert. The flight crew then decided to perform an air turnback.

Narrative

After takeoff; as directed; I raised the gear. Subsequently; the nose and right main gear lights did not extinguish. At or about the same time; we got the alert HYD 3 FAIL. The Captain immediately set the speed to 215 to avoid overspeeding the gear and flaps; still at 10 degrees. The hydraulic fluid quantity in Number Three was indicating zero. Once we contacted ZZZ Departure I requested priority handling and requested vectors and an altitude to keep us clear of traffic. We told them we needed time to troubleshoot our hydraulic problem. We then ran the checklist for HYD 3 FAIL in the QRH. The big picture was we were required to do a manual gear extension; land with flaps 35; no autobrakes; and nose wheel steering control was limited to 25 degrees to the right. Together; we decided landing on [Runway] XXC and stopping straight ahead on the runway was the best course of action. We requested extended vectors for the ILS to [Runway] XXC. We configured according to the checklist. Landing was uneventful. We stopped straight ahead on the runway as planned. Emergency crews responded and indicated they saw smoke coming from the right main gear. The brake temperatures were normal and deduced the smoke was from dripping hydraulic fluid coming from the right wheel well. Once it was cleaned up; Operations towed us back to the gate. Cause - hydraulic line in the right wheel well for the Number Three Hydraulic System severed; causing loss of fluid.