Ross Hassig reports that the sling was used in Mesoamerica although it does not appear in works of art.
Ross Hassig, War and Society in Ancient Mesoamerica (Berkley: University of California Press, 1992), 31
Slings do not appear in works of art and were probably low-status weapons, as they were in medieval Europe, because of the utilitarian skills they demanded. Nevertheless, the distribution of slingstones throughout Mesoamerica indicates the continued use of this effective and inexpensive projectile weapon. Standardizing the slingstones by sizing pottery or grinding stones made slings more effective, in that each slinger would then know with relative certainty how far and accurately he could throw. However, their weight limited the number of slingstones that could be carried long distance, so unless they would be retrieved for reuse—and the prevalence of fragments suggests that the ceramic ones were relatively fragile—slings were probably used primarily to defend settlements where transport was not an issue.