Why is a refund not showing on a customer's bank statement?

After you initiate a refund, Stripe submits refund requests to your customer’s bank or card issuer. Your customer sees the refund as a credit approximately 5-10 business days later, depending upon the bank. A refund might not be visible to the customer for several reasons. The most likely reason is because a refund issued shortly after the original charge may appear in the form of a reversal instead of a refund. In the case of a reversal, the original charge drops off the customer’s statement, and a separate credit isn’t issued.

Login to your Stripe dashboard, find the payment in your "Payments" view and then filter on refunds. Click the refunded payment to see more information.

In the Timeline, click the "View details" link to see more information on the refund:

You can then see more details about the refund. In this case it went through as a reversal and therefore the initial charge and refund will not show on the customer's bank statement:

In rare circumstances, refunds can fail if the customer’s bank or card issuer has been unable to process it correctly. The bank returns the refunded amount to Stripe and they then add it back to your Stripe account balance. This process can take up to 30 days from requesting the refund.