goatskin

There is also no issues with drawing breath as suggested by others as the goatskin is under the same water pressure as his lungs.

You would need a tremendous amount of extra weight to submerge a goatskin full of air.

You cannot dive with an inflated goatskin, unless you carry extra weight, like stones.

Why have an inflatable goatskin that you're inhaling from for swimming on the surface?

The swimmer without a goatskin looked to me to have an arrow (or two?) in them that would have come from that direction.

The swimmer isn't inhaling, he's blowing into the goatskin to keep it inflated.

I spent an afternoon visiting peaceful island monasteries, where the monks cheerfully handed me 500-year-old goatskin Bibles to leaf through.

In this image the 'goatskin' users appear fully clothed in dresses (could they be scale armour or other weighted clothing), only one has a beard.

Maybe the goatskin has rocks in it too, to balance buoyancy?

That goatskin is like 20-30 liters - good for several minutes of breathing and those 20-30kg of extra floatation allow for the typical armor/chainmail and weapons.

I also don't think an inflatable goatskin has enough air capacity to be relevant.

Take a look at the antique bond on goatskin parchment that financed a whole dike, now paying a mere €11.35 a year, valued only as a curiosity.

Their tongues were often coated in ice and the dogs swelled to resemble "an inflated goatskin bag," the authors wrote.

A person without a goatskin appears "above" them (could be behind depending on perspective) and IMO appears to be getting shot with arrows.

"Parchment" can also be made of calfskin or goatskin.