I have a number of family members and friends that are not US citizens, live overseas and have US bank accounts. (Citi, BoA, Chase). No doubt it can be done. I know USAA will (my son just did) but you need to be eligible. Citi as well – if you can stand them. PenFed might (not a bank but a credit union).
Overseas offices of US banks (like Citi) are separate legal entities and require accounts domiciled in each country. For most, however, transferring funds between offices is straightforward (but no less costly). Account management from foreign offices is usually allowed only for large ($$) customers.
US banks have been limiting foreign based accountholders (regardless of citizenship) because there has been a great deal of fraud. They have also curtailed direct online international transfers, for the same reason. Most still transfer but require phone interaction with a bank operator.
just to clarify: US banks will withhold taxes on interest and dividends paid to non-US citizens regardless of domicile. Citizenship is determined when the account is opened. Accounts opened by US citizens with overseas addresses are not subject to withholding. Wire transfers are not subject to withholding regardless of owner citizenship, just fees.