Well this sounds like a big hot mess and I can see why u thought it would be ok considering this girl was supposed to be a friend, unfortunately when money exchanged hands it became a business transaction. You provided services for cash which changes things a lot. Despite her being a mate you should have gone through all your consultation, after care and signing paperwork where necessary. I spray tanned my own cousin and her friends in the summer but they still were asked to fill out the consultation sheet with me and sign it to say they understood everything and that's "just a tan" there are no friends in business in sorry.
Next thing you really need to do is seek legal advice immediately, you were not insured I know, but you still need to find out where you stand and how to proceed. It might be beneficial to block the mother from contacting you and tagging you in posts on social media. You may have a case for slander but in the mean time, stop texting and if anyone approaches you aggressively contact the police. It might be a good idea to only communicate through solicitors of that's how it's going to go. And in future purchase insurance if you intend to do any work on a mobile basis. I hope this gets resolved ASAP. Your client is wrong for being abusive and threatening and that's a matter for the police. Good Luck