Are there countries with free banking?
I'm going to stick my neck out and say that the answer is no, not because I have a specific reference (people tend not to publish papers stating that something doesn't exist) but monetary systems has been a keen interest of mine for the past decade and if free banking was currently being practised somewhere I would almost certainly know about it.

It has however been practised historically, perhaps most notably in Scotland 1716 - 1845.

You can find many proponents of free banking in the Austrian School of economics, for example George Selgin.