This issue has been going on for a couple of weeks now (at the least). I recommend contacting Etsy support and ask them what they intend to do about it. Their bug (of sending these Out of Stock emails to the last buyer) is now costing sellers. Luckily for me, my buyers contacted me first to ask why it was out of stock since they just purchased it. I'm sorry you have been caught out by Etsy's incompetence.
The item was not unavailable. She bought the item which if you had the order, you could have responded back that you had the order and was shipping it if she asked right away about the email.
This is a glitch that has reappeared. The item is for sale when the buyer purchases and then some old programming is cutting in and issuing a sold out email when it shouldn't. It will get fixed eventually.
If she wishes to return the item then I would accept the return and refund her. If you are in the US send her a prepaid return label.
This is another reason to have self insurance fund set up for glitches like this. In the computer age it is going to happen and any seller needs to be prepared. You can contact Etsy, explain the situation and ask them to consider reimbursing the buyer. They just might.