Emergency services were alerted to a collision on Jenkin Road, Wincobank, at 3.30pm on Tuesday and found an 11-year-old boy with serious injuries.
He had been struck by a grey Toyota Avensis.
Last night they boy remained in hospital in a critical condition.
A 24-year-old Sheffield man was arrested on suspicion of driving under the influence of drink or drugs and failing to stop at the scene of an accident.
He was later released under investigation.
Witnesses or anyone with information should call South Yorkshire Police on 101 and quote incident number 509 of September 24.