The child was taken away from the 35-year-old mother who slapped her, then swore at and pushed a woman who intervened.
The mother was at Derby Bus Station on Wednesday evening when she was seen slapping the child.
Police initially arrested the mother on suspicion of common assault, interviewed her and gave her two cautions.
Derbyshire Police said the child was now being looked after by social care.
It is legal for a parent to physically punish a child in the UK if the punishment is "reasonable".
Scotland and Northern Ireland have similar laws.
A United Nations report suggested the UK should pass laws to ban parents from smacking their children.