The modern day Balearic sling apparently originates from the Iberian peninsula, but the design very likely dates to ancient times. The statue of David by Bernini from 1623-4 holds a sling very recognisable as what we would call today as a "Balearic" sling. There is also a Spanish painting from about ~1440 clearly depicting a split pouch, and likely similar in design. If the design dates back that far, I think its extremely likely that something similar existed in ancient times.
However, I believe the specific sling designs that directly originate from the Balearic Islands have been lost to history, and were gradually replaced in favour of the design people typically use on the islands today.
That is, a sling with a braided from loose fibre with a finger-loop that merges into flat braided retention cord (typically un-tapered or slightly so) a split pouch of 2 or 3 thongs growing in thickness (2 is most common), a thick belly that quickly tapers to retention cord thickness, then gradually reducing to the thickness of the finger-loop or thinner, and ending in a cord grip with tassel (no release knot). Like I say, there is some variation, but these general features are what make a Balearic sling distinguishable from other designs.
MMF wrote on Nov 13
th, 2022 at 10:13am:
They made these slings from esparto, horse hair, sinew, or black-tufted rush. Basically any material they could get there hands on. So, any sling from the Roman era has rotted out of existence.
Esparto and black tufted rush are two names for the same thing. It's a particularly excellent material for this style of sling, for numerous reasons.