Only you can guage if she is lying or not by the way she has contacted you and advised you of the problem.
I'm not sure what year the iPhone 4 came out, but could it be that the battery is approaching the end of it's life?
If she has paid by PayPal, then chances are you may have to give a refund, regardless of whether the phone is working or not, depending on how she raises a complaint with eBay.
If it was me, I'd keep it professional and ask her to send everything back to you (ensure she knows she has to send it back with tracking). Advise her that once it is received and checked, you will refund her the full amount including the return postage costs if it is indeed faulty.
If you cannot find the fault, than it is up to you how you proceed from there, although I would probably refund the full amount (excluding the postage costs) and knock a little bit off tha amount to cover eBay / PayPal fees.
