Sport shoes outsole material

I am working on a footwear project and i am looking for an outsole material that has the handfeel and texture similar to a “antistress ball” or a silicon gel insole (ie : as very soft and elastic as skin) but with a very high resistance to puncture and abrasion (as the outsole will be very thin).
I can’t find the suitable matierial yet.
Thanks a lot for your ideas and advises.

Low durometer TPE type materials are the most similar feeling, anything skin like with gel inside will rip in the first hour of use.

Hytrel and Kraton are common ones, made with mixing plastics with oil. Which results in some “interesting” (read horrifying) results after a few years after molding when attempting to self formulate.

thanks a lot nxakt for your help. I will check these options.
if anyone has different idea, you are welcome. I d like to explore different paths.

someone also recommended to use latex rubber for the outsole ? does anyone knows the pros and cons of the option for a sport shoe with a extra slim outsole ?