This appears to be an urban legend since in the 2007 Arkansas Law Notes which contains "Reports to the Arkansas Bar by the Faculty of the University of Arkansas School of Law" there is an article that runs from page 113 to page 128 titled "Virtually Legal: Or don’t believe everything you see on the internet!" by Lorraine Lorne who was the Assistant Director of Law Library at the time of the articles writing.
The article is fairly long but makes for an interesting read but the relevant part comes towards the end,
Finally we return to consider, and conclude with, the stupid law with
which this investigation began.
It is legal in Arkansas for a man to beat his wife no more than once a month with a stick three inches or wider . . . on the courthouse
steps . . . on the state-house steps. . . .
Although gleefully included on almost all of the stupid laws websites,
no such law was found in any of the various compilations of Arkansas
statutes. However, after checking the various stupid/dumb/crazy laws
web-sites again, the states of Alabama, Arizona, California, South
Carolina appear to permit beating one's spouse within certain limited
circumstances: once a month, or only on Sundays, on the courthouse
steps or the State House steps, with a leather strap or a stick no
wider or larger than three inches or one's thumb, with the permission
of the victim!
An interesting aspect of researching this particular "law" is that I
first asked a criminal law professor if he had ever heard it. He said
"Of course. My criminal law professor told us that this legal concept
dates back hundreds of years and is usually known as the"rule of
thumb"! Wanting to follow up on this nugget of knowledge, I looked in
Black’s Law Dictionary and found no entry. The Oxford English
Dictionary defines rule of thumb: a method or procedure derived
entirely from practice or experience, without any basis in scientific
knowledge; a roughly practical method. Finally I searched one online
legal periodical index for possibly relevant articles; there were over
6000 hits. Since an in-depth consideration of the rule of thumb is
beyond the scope of this article, I'll leave that for the reader to
delve into. However, one author wrote an interesting article that
suggests that "rule of thumb" has nothing to do with punishment and
very little to do with thumbs.