The five-year-old was injured when the white Range Rover Evoque she was travelling in was involved in a head-on collision with blue Citroen C3 being followed by South Yorkshire Police on Fish Dam Lane, near Royston, Barnsley.
A police pursuit was mounted when a police officer spotted that the Citroen was travelling with a smashed windscreen and signalled for the driver to pull over.
The motorist refused to stop and crashed into the Range Rover while trying to lose the chasing police car.
A 29-year-old man in the Citroen was seriously injured and taken to hospital.
He was arrested on suspicion of burglary and causing serious injury by dangerous driving.
The Citroen he was driving was stolen in a burglary in Barnsley.
Anyone with information should call 101 and quote incident number 419 of February 28.