We are in Washington state.

My daughter is visiting me for the summer and I would like to try for custody. If her mother is a permanent resident of Sweden what recousre does she have if I refuse to let my daughter go back at the end of the summer? What can I do to keep her in the states and with me?

Her mother has sole custody and aloud her to visit me. The case started in Pennsylvania but neither of us live there anymore. Does Washington state have a possesion law for children? I just want to do whatever I can to keep her in the states.

What you are contemplating is international child abduction by a parent. Many countries have a reciprocal agreement with the US (and other countries) for child support and child custody. Sweden is one of them. You cannot just "keep" her. If you want to modify custody, you will have to file in court for modification. You will have to show cause as to why it is in your child's best interest to remain with you and that there has been a significant change in your child's living condition to warrant a change of the custody order. In short, NO you cannot refuse to send her home. You really don't want to start this fight. If you keep your child, you have destroyed ANY chance of ever getting custody (and risk losing visitation rights) in the future.