If you want only an ATM card and not a debit card, you might be able to get one. Banks don't advertise it, but I got an ATM only card from Bank of America.
An ATM only card lacks the Visa or Mastercard logo. It can only be used on interbank networks like NYCE PLUS STAR. Or maybe only at that bank’s branches. You always have to key in your PIN and there is often withdrawal limit of $500/day because a debit card and PIN is immediate cash.