Residency or DNI is not required by all banks. You just have to find a bank willing to work with you. I helped a friend in November get an account at Banco Frances.
As a non-resident foreigner, the will likely ask you for your passport, a CDI number (tax payer ID), which you can obtain at many AFIP office (same day), and one or two utility bills on your name. They will only offer you a savings account, and may sign you up for home insurance with the account, which you can cancel after the first month. They will also want a small deposit to open the account (few hundred pesos).