Hello!
Can an account with a non-resident alien have only a foreign address on file? The NRA used to live in the US, now lives in Taiwan, and has mail sent to the branch here in the US. Doesn't the NRA have to have a US address on file?

No. There are US banks that open accounts for individuals outside the US who have no US residence. It is a question of your risk appetite, controls, etc. re deciding what you are willing to accept. There are various reasons banks do this.

Your W8, if it is an interest bearing account, has to have his foreign address.

Other than that, his mailing address can be foreign or domestic.

Some NRAs own homes in the US so maintain a local address.
Others just come here on vacation and keep their foreign address or use Hold Mail and pick up their mail when they come.

Make sure your BSA Program, especially CIP, covers this type of situation if you keep the account.

IMO, mailing statements to the branch is a stupid idea. He can pick up 24 months of statements at a time, then successfully dispute every charge on them, because you didn't provide his statements to him until he picked them up, right?

Hold Mail is a bank service for which the customer pays a fee, and there is a Hold Mail contract that covers such scenarios.

A customer cannot just choose to have his statement mailed to the bank.