You can certainly file a personal injury lawsuit and, depending on how much, you might even be able to use small claims court.
However, you might not have to file a lawsuit.
There's a couple of things to do first.
1 - Has the owner been identified? That's the first thig you have to do. If you can't do that, there's nobody to file against.
2 - Document the incident. Get a copy of the report from Animal Contral. Get dated photos of the injuries. Get your child to a doctor to get the injuries treated. Without treatment and medical expenses there is little or no claim.
3 - Your child's emotional reaction to the dog is not worth any money. But for her own sake, a couple of visits to a child psychologist to overcome any fear of dogs should be done ASAP. The sooner she gets seen for it, the sooner it will be relieved.
4 - Put the dog owner on written notice of the incident and the injury. That is likely to get reported to the owner's insurance company which, in turn, will probably offer something just to make the claim go away.
5 - Study up on CA dog bite law.
http://www.dogbitelaw....