Ross Hassig notes that the Maya word for "sling" dates as far back at 1000 B.C.
Ross Hassig, War and Society in Ancient Mesoamerica (Berkley: University of California Press, 1992), 205n51
The antiquity of slings is based on linguistic evidence, with the Maya word for sling going back as far as 1,000 B.C. (James A. Fox, personal communication). Blowguns may have been used as well, but not as military weapons (Shook 1946)