Are there countries where non-residents can open bank accounts without visiting in person?

In some country? Yes.

Be informed that in countries like the US, Canada, UK, Ireland, European Union, Japan, South Korea, Australia, New Zealand, etc. you simply cannot open a remote bank account.

You can open pseudo bank accounts with reserve trust companies or e-money payment institutions, but these would not be a proper bank account, but more or less a wallet account.

The only solid option that I know of is off-shore banks, as that is exactly what there are designed to do, open accounts for foreigners, without them having to visit the country itself. There are minimum requirements on the amount of money you have to maintain, but it is a viable option nonetheless.

Most offshore banks will provide USD, GBP, EUR clearing accounts.