PA28 pilot reported the engine failed during initial climb. Pilot made an immediate turn back to airport and due to landing long was unable to stop aircraft on runway; resulting in runway excursion.

Narrative

When departing ZZZ from Runway XX; I experienced the partial loss of engine power initially and quickly after experienced complete engine failure at 600 AGL. The reason for the engine failure is still unknown. I elected to turn around and land on Runway XY as I deemed it was; without doubt; the safest option for me; the public; and for the aircraft. I encountered a tailwind on my short approach and eventually touched down halfway down the runway at roughly 70-75 kts. I knew I would be safer and have better control of slowing the aircraft down by braking on the runway; even if it meant touching down early and fast. I eventually had to completely lock my brakes and ultimately stopped 100 ft. beyond the end of the runway on the 'extended center-line' in a flat grassy area. I did not damage any airport property; however; the aircraft did blow its front tire. This was the only damage done in the entire event. I was solo; and this was my first real emergency situation. I followed all procedures of the flight exactly by the checklist and am fully certain my actions did not cause the engine to shut down. The aircraft checked out great in pre-flight and the engine run-up was great and within tolerances. My hope is that Maintenance can uncover the cause of the engine failure. Ultimately; I feel that given the circumstance of a low-altitude engine failure and considering the environment around me; I made the best possible decision to land where I did and believe it provided the safest possible outcome available.