*(I was hoping somebody else would answer this, because function fields are not really my area and I hoped I would learn something from the answer; but nobody seems to be biting, so...)*

Iwasawa theory over function fields definitely exists, and in many ways it's easier than number-field Iwasawa theory -- there are *more* nice tools available, such as the Grothendieck--Lefschetz trace formula, which aren't there in the number field setting.

For instance, [here][1] is a paper of Goss and Sinnott from the 1980s which (among many other results) proves an analogue of Herbrand--Ribet for the class groups of function field extensions arising from Drinfeld modules. 

  [1]: https://0-projecteuclid-org.pugwash.lib.warwick.ac.uk/download/pdf_1/euclid.dmj/1077304443