There is now a deep analogue for function fields, the geometric Langlands program pioneered by Beilinson and Drinfeld. See there for more details.
A good survey is in
Mark Goresky, Langlands’ conjectures for physicists (pdf)
(which however, despite the title, has nothing in it that is particularly related to physics).
Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219; Edward Frenkel, Commentary on “An elementary introduction to the Langlands Program” by Steven Gelbart, Bull. Amer. Math. Soc. 48 (2011), 513-515, abstract, pdf