How about Kevlar? This is an organic compound and could conceivably be produced in an animal equipped with the right biochemistry. Another component that might be used instead of, or as well as Kevlar, is spider silk or a material similar to it.
Spider silk is elastic and very strong although normally encountered in very fine filaments that are easily broken; it can be spun and made into much larger and more resilient objects. For instance this very rare rug made entirely out of Madagascan golden orb spider silk:
https://www.wired.com/2009/09/spider-silk/
Update it would seem that some spider silk is even stronger than kevlar https://www.wired.com/2010/09/super-strong-spider-silk/
The size of some of the webs has to be seen to be believed. Made me feel rather quesy walking under one of the webs at dusk even though it was high up between trees.
