The child was attacked in her pushchair after her mother stopped to open a garden gate in Barclaugh Drive, Coylton, at about 10:30 on Monday.
The small black dog appeared from woods, known locally as the Sundrum area, and bit the child before running off into a neighbouring garden.
She has since been discharged from Glasgow's Royal Hospital for Children.
The girl was transferred there after initially being taken to Ayr Hospital.
PC Gerard MacLean, from Ayr Police Office, said: "Officers have been carrying out inquiries in the local area to try and establish where this dog has come from.
"The dog did not have a collar on at the time and was not in the company of its owner.
"I would appeal to anyone who either witnessed the incident or has seen a small black dog in the area to get in touch."
A 10-month old baby girl has been treated in hospital after being bitten on the face by a dog in South Ayrshire.