The woman was taken to the Western Medical Center in extremely critical condition following the incident.
Investigators have been unable to determine who the woman is, and she has been unable to communicate due to her injuries.
The woman was pushing a shopping cart containing her belongings at the time of the accident, leading police to theorize that she might be homeless.
"But until we find out who she is, we will not know that," Cpl. Anthony Bertagna, a spokesman for the Santa Ana police department said.
The woman's fingerprints have been compared to local, state and federal databases, but no matches were found, the Los Angeles Times reported.
She also does not match any missing persons descriptions in Santa Ana.
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Cyclist struck and dragged underneath car