I don't know how this kind of country is called, the names of those who play it, but I know I need more!
According to Wikipedia, the entire soundtrack was inspired by gospel blues pioneer Blind Willie Johnson's Dark Was the Night, Cold Was the Ground, which was one of only 27 songs launched into space aboard Voyager in 1977.
The main instrument is called "slide guitar," if you search for that, you might find some other work that you enjoy.
Ry Cooder's page on IMDB.com (the Internet Movie Data Base) lists 22 motion pictures and television shows where he was the composer and performer. Scroll down and click on the "Composer" tab.
Paris, Texas, from 1984, was his fourth film score.