My advice that has 100% worked and protected me as a seller is as soon as you receive an email from a buyer about something like this is get on the phone with an eBay rep so they can document everything. Have them go through and read all the emails between you and the buyer and ask the rep to make notes, this helps if/when you have to contact ebay again about this issue if things progress.
How this just recently helped and covered me. I recently sold a brand new in the box string trimmer. Buyer emails me around 2 weeks after saying there is something wrong with it but very vague, i respond back super nice asking for more info, where they respond its the starter part and they have an estimate for the fix and wanted money back. Thankfully I worded my questions in a way that they responded that they did in fact use it, there exact words were "it worked initially, not now". My thinking is its a string trimmer, you cut grass with it, i have no idea how the buyer used it or how it broke, if it even did break. I have my suspicions they just saw it for a lower price and wanted money back, but that's purely speculation on my part(the ebay reps also agreed with me and told me so). So after the buyer told me they used it and that it did work but now doesnt I immedialty contacted ebay, spoke with a rep on the phone, be super nice to the reps, ask them to pull up and read all the emails between you and they buyer which they can, in my case the rep say the buyer admitted when they received it it was in fact working so she said dont worry i fulfilled my ebay obligations and im covered if it escalates, she even wrote notes down for the next rep if i have to call again(actually ebay calls you back). i kindly inform the buyer theres nothing I can do, i quoted that they said it was working. needless to say they werent happy, they wanted me to pay for a replacement part and the cost for a repair shop which actually came out to more $ than I paid for the trimmer, buyer claimed ebay has a 30 day guarantee against "faulty" items . After that email i immediately contacted ebay again, this time the rep made it seem that i was on the hook if the buyer broke an item within 30 days and I would have to accept a return, she actually mentioned walmart and if someone bought something from walmart and it broke within 30 days they would take it back, but after i expressed my view telling her theres no way im remotely like a walmart and we went through the emails again she came around to my side and she agreed the buyer is probably trying to pull a fast one, again be super nice with the reps talk to them like they are your friend and it will go much better. So her advice was for me to kindly say since they used the item and it broke while they were using it that theres nothing I can do. I then responded to the buyer kindly asking them to forward the written estimate and the name of the repair shop they supposedly brought it to. They didnt email back and instead opened a case and wanted a full refund. So, you guessed it i immediately contacted ebay on the phone and had that rep go through all the emails and the notes the previous reps wrote and she agreed the buyer used the item and it was working and it broke when they were using it, she said she will close the case and there will be no more action taken against me. Instead she said ebay will cover the return, which they did. Did not cost me anything. unfortunately the buyer got the full refund and to keep the item. This also worked when a buyer wanted to return an item claiming it was damaged, but the part was a solid metal machined part and there is no possible way it could have been damaged but I accepted the return anyways, when i recived the part back there was zero damage anywhere on it, so i took a bunch of photos and wrote back to the buyer asking if they can kindly point out what was exactly damaged, they never responded so i contacted ebay onthe phone and told them the story and had them read all the emails and they sided with me and refunded all my money and i got to keep the item, so i actually kept the money the buyer paid, got reimbursed all shipping costs and kept the item to resell again.
I skimmed through some of the replies and some are wrong, if you fulfill your obligations then you are not on the hook. But it will take some effort, some phone calls, patience and get everything documented.
In your case i would have called ebay immediatly after receiving that 1st email, talk to the rep on the phone not through chat, have them read the emails and see that they buyer is not claiming anything wrong with the item but instead they cant afford it which is not covered under ebay policy. Have the rep write notes incase you have to contact them again it will be another rep. This will cover you in case the buyer opens a case and they give a different reason, they will look dishonest. This is where being nice to the reps come in, get them on your side and understand your not a walmart and you cannot afford even to pay for return shipping. This only works if you really are not at fault, that you did everything by the book.
I read tons of posts from sellers saying ebay is only on the buyers side and will side with them no matter what, which is not the case. Ebay does care about sellers and dont want them ripped off. But the thing is ebay is not going to step in and help you unless you ask them to. Most of the transactions are automated, thats why talking to an actual person is needed.
