The girl was bitten “a number of times” by a Jack Russell-type dog in Skegness on Saturday afternoon, Lincolnshire Police said.
The owner allegedly refused to give her details to the girl’s parents after the incident off Grand Parade.
Police have now released a picture of a woman who is believed to be the woman’s owner in bid to track her down.
The dog was apparently sitting under the owner’s chair on the beach and was not on a lead.
Officers are treating the incident as an offence of having a dog dangerously out of control in a public place.