There is a town with that name in Tipperary, Carrickbeg.
https://en.wikipedia.org/wiki/CarrickbegThe difference is that the Irish for that uses the article "an" (the) before it so it comes out a little differently to what you have: An Charraig Bheag. But I don't think it makes a difference whether you use the article or not.
Then there is a village not far from there called Carrigeen.
https://en.wikipedia.org/wiki/CarrigeenSo either would work. It's up to you which one to choose.
As for Arkansas, it would probably remain the same in Irish, or changed to fit the orthography of the Irish language: [Árcanso?] (Please don't use this without getting further input on orthography and spelling.) Often, place names like this are not translatable. Again, I refer to Wikipedia University and the following quote:
"The name Arkansas initially applied to the Arkansas River. It derives from a French term, Arcansas, their plural term for their transliteration of akansa, an Algonquian term for the Quapaw people . . . Akansa is likely also the root term for Kansas, which was named after the related Kaw people."
There may be translations from those languages for the names of those people but getting into a "translation" into Irish starts to border on the ridiculous. But there is one interesting comment about the capital of Kansas, Topeka:
"The name of Topeka, capital city of Kansas, is said to be the Kaw word Tó Ppí Kˀé meaning 'a good place to grow potatoes'."
Get more input on this before you decide anything permanent.
Cheers,
Tim