If this was a valid law, it should be possible to point to a successful prosecution made under its provisions. There hasn't been one.
The reason this myth has grown legs is because of an article introduced into the Arkansas legal code in 1881. The article in question, which I have reproduced below, is taken from Lexisnexis.
1-4-105. Pronunciation of state name.
Whereas, confusion of practice has arisen in the pronunciation of the name of our state and it is deemed important that the true pronunciation should be determined for use in oral official proceedings.
And, whereas, the matter has been thoroughly investigated by the State Historical Society and the Eclectic Society of Little Rock, which have agreed upon the correct pronunciation as derived from history and the early usage of the American immigrants.
Be it therefore resolved by both houses of the General Assembly, that the only true pronunciation of the name of the state, in the opinion of this body, is that received by the French from the native Indians and committed to writing in the French word representing the sound. It should be pronounced in three (3) syllables, with the final "s" silent, the "a" in each syllable with the Italian sound, and the accent on the first and last syllables. The pronunciation with the accent on the second syllable with the sound of "a" in "man" and the sounding of the terminal "s" is an innovation to be discouraged.
HISTORY: Concurrent Resolution No. 4, Acts 1881, p. 216; C. & M. Dig., § 9181a; Pope's Dig., § 11867; A.S.A. 1947, § 5-102.
Note that the provision in the code merely describes the unfavoured pronunciation as "an innovation to be discouraged". That doesn't make that pronunciation illegal — it merely states the official opinion and preference of the state legislature at the time this article was included in the Arkansas legal code.
It may also be of interest that in the neighbouring state of Kansas (where I actually live), there are several place names that include the word Arkansas, like Arkansas City. Here, those place names are actually pronounced as written (though Arkansas City is more often than not simply shortened to 'Ark City' in speech, which renders the pronunciation of the end of the word a moot point).