The girl was struck by a black Volkswagen Golf convertible on Holly Lane, Erdington, just before 5.30pm.
She was taken to hospital and remains in a critical condition, while police appeal for witnesses of the incident.
A 47-year-old man was yesterday arrested on suspicion of causing serious injury by dangerous driving and has since been released under investigation while enquiries continue.
A spokesperson for West Midlands Police said: "We're still trying to trace the car which we believe sustained significant damage to its windscreen and passenger side during the collision.
"We're keen to hear from anyone who has information about the collision - or has seen a convertible with noticeable damage over the last few days.
"You can contact investigators directly at SCIU@westmidlands.police.uk or via Live Chat on our website, quoting log 3239 of 27 May."
