Under questioning, the woman told investigators her name was Joanna and that she was a mermaid.
According to The Independent, a physical examination revealed the woman suffered from webbed feet, a relatively rare condition in which some or all of the toes are either fused or feature a loose flap of skin between each digit.
Officers described her behaviour as "bizarre" and said they were still trying to establish whether her failure to state her identity was a result of medical reasons or simply a decision not to.
The woman was "coherent and responsive" but unable to answer most questions, hospital sources told The Fresno Bee.
An initial appeal described the woman as "aged between 16 and 30, with brown hair, brown eyes and webbed toes on both feet". She was not carrying a driver's licence or ID card.
Police later released a statement saying they now believe the woman is a 33-year-old from Virginia who travelled to Fresno last week to check out the area with a view to moving there.
The Fresno County Sheriff's Office did not release her name on Thursday, adding she may have been the victim of a crime, according to the Bee.