Customarily, you are responsible for it if you do not have any policy posted prior to the order taking place. You offered the free shipping. The buyer can apply for PayPal return shipping reimbursement if they registered for it in their PayPal account. A dispute is not required for that but terms apply.
Now, you can issue a refund but deduct the shipping costs from the amount of refund but may risk angering the buyer.
In a dispute, the amount the buyer paid will be refunded to the buyer if PayPal is involved with resolving the matter. You can also negotiate with the buyer through a dispute for a partial refund to see if they'll agree. If not and they escalate to PayPal, it means you have no deal and the customer is looking to get entire amount paid back.
If you have a return policy, stating that shipping costs will be deducted on returns on free shipping orders, PayPal may take that into account as part of your evidence but no guarantees.