Hell, I might have to give this a go.

I've been meaning to do some felting for certain ancient/medieval hats, and if I ever get around to it I would certainly have off-cuts that would make a convenient sling pouch.

My felt is elastic. Perhaps, there are different qualities. So, my feeling is a sensation of a soft throw. Pleasant.
Others tests are necessary. I'll spin wool, felt the strands and plait a "balearic" sling.
A long time ! 

I have made a few wool slings from both twisted yarn, and felting yarn, and they work very well. However, there is one vital thing to bear in mind, which is that these slings are only good if they have been heavily stretched while they dry. I do this by soaking the finished sling in water, and then tie the release cord to a bucket full of heavy things, and lift the bucket and sling by the finger loop, which I slip over a hook on the garage ceiling.
The whole sling is then evenly stretched, and once dry, the wool sling has similar elasticity to rope (no perceivable stretch).
The heavier the weight used to stretch it, the less elastic it will be.

Thank you Lithera.
I'll do like you say.

my spinned wool is not very elastic but after I felt the braid, the wool is very elastic.
So, I will test the sling 2 times. Before I will stretch the sling and after to feel the difference.

Hello,
If you search : fulling on Wikipédia, you find an article about the treatment of wool in history.
To do my sling,I make strands with hot water and urine.
The video is to show the result with my wool.

Recently performed some more experiments with wool slings, and I found a way to make them work really well.
First, tightly braid a wool sling using yarn. (I like 2 ply twist as its fast and can be done tightly)
Once complete, stretch it in the way I described above.
The next step is to dip the whole thing into liquid beeswax, take it out, and rub off all of the excess beeswax with a glove. I then like to stretch it again before the beeswax cools so that the cooled beeswax prevents the wool from 'relaxing' back.
The resulting sling is very abrasive resistant, quite light, inelastic, flexible, comfortable, 'grippy' (no release knot needed), and its handling characteristics don't change even if dunked in water.
Wool is now my favourite sling material when made this way.