by contacting local authorities, i assume you mean you filed a police report for ID theft?
you also should have an alert on your credit report showing that this person has stolen your ID.
if she got a credit card in your name, the billing would be mailed to a different address than yours? or if the billing is mailed to you for payment and you saw charges you did not make, you would have a record of calling the credit card company and disputing the charges.
i suggest you provide copies of the police report and other authorities you have contacted showing you are not the person they are trying to locate.
go to freeannualreport.com to get copy of your credit report. black out all personal info except the last 4 digits.
ie; social security number
credit card numbers;
so say she used XYZ credit card with the last 4 digits of 1234, there should not be any card under your name for the same.
they have to prove that you were the one who rented the car.
it they don't back off, then i suggest you contact a lawyer for your legal rights.
good luck