Elastic properties of materials at low temperature It is common knowledge that materials are more brittle at low temperature. But does it apply also on elastic deformations or is it just matter of plastic deformations? 
Practically: Is it possible to make a flexible cloth, rope or string working at $\approx 0 K$ 
( e.g. from whiskers of pure metal or ceramic ? Or other monocrystal. )   

My reasoning:
From fundamental physics/chemistry point of view it is clear that plastic deformations which involve reorganization of chemical bonds is hampered by low temperature simply due to Arrhenius law. The easy modes of deformation and self-healing are kinetically prohibited which leads to catastrophic failure.
But I don't see why this should apply to elastic deformations of e.g. some metallic or ceramic crystals where all atoms only slightly deviate (~few pico-meters) from its equilibrium position in crystal.
Rubber is different since its elasticity has entropic origin and involve dynamics of conformational changes of polymer chains, which are affected by Arrhenius law as well.

EDIT: by $\approx 0 K$ I meant something technically reasonable, like few kelvins, liquid helium temperature. I didn't mean some super low teperatures where fancy things (like Bose–Einstein condensation happends )
 A: I can't give a very theoretically driven answer, just one based on personal experience.  If you limit yourself to the few Kelvin range (say 4K to 10K, as noted in your edit) then  materials like Berillium-Copper retain much of their elasticity. For example a BeCu spring remains perfectly functional at ~10K.  It will compress and expand when loaded and unloaded, and vibrate/ring when perturbed (Low temperature STMs commonly use spring suspensions to dampen vibrations at operating temperatures down to 4K).  I've also made flexible copper rope (~1.5" in length) by braiding bundles of ~100um diameter wire. They remain reasonably flexible down to 4K and are used to cool a sample stage that moves around relative to a cryostat.
A: This is only one example, but in their paper "Mechanical properties of carbon nanotube fibers at extreme temperatures," Zhang, C. $\textit{et al.}$ subjected carbon nanotubes to very low temperatures and found the follow result:
"The mechanical strength and Young’s modulus of the CNT fibers decreased with the increase in testing temperatures. This behavior is a result of the temperature dependence of the carbon bond strength and intertube interactions"
The rest of their findings seem to indicate that carbon nanotubes get stronger as they get colder (down to -196$^{\circ}$C or 77K, the temperature of liquid nitrogen) but lose "flexibility" broadly speaking, only gaining tensile strength in a relatively linear way. Most options to make a more "flexible" rope that is resistant to low temperatures would probably be organic-based, so this might be a good indication that you will always give up flexibility in the rotational axis at low temperatures even if you are gaining tensile strength along the longitudinal axis.
https://pubs.rsc.org/en/content/articlelanding/2019/nr/c8nr09637f#!divAbstract
A: Upon being subjected to stress, a sample of material will stretch elastically first, and only if stretched beyond its elastic limit will it deform plastically. 
Plastic deformation is the (mostly) irreversible movement of atoms and crystalline defects within the material which, as you point out, will be modulated by thermal activation- that is, there are activation energies associated with plasticity mechanisms, and if that activation energy is present thermally, then that deformation mechanism will be active. 
The elastic portion of the stress response is established by the nature of the bonding between atoms in the bulk and as such is not subject to thermal activation of, for example, diffusion processes in the solid state. 
By this reasoning, we would not expect that reducing the temperature of a sample would have a first-order effect on its elastic modulus.
A: My answer is No, and I believe this explains the issue of any flexibility.
Even when water freezes, the molecules are still vibrating. Only at absolute zero do molecules stop moving completely. What sort of phase change would this cause? According to Einstein, who based his work on an Indian physicist Satyendra Nath Bose, this would result in a completely new form of matter. 
Albert Einstein speculated that when molecules stop moving, the atoms would fall together and merge into one atom. The individual atoms (oxygen and hydrogen, in the case of water) would loose their identities and form what Einstein called a "superatom", or what we now call a Bose-Einstein Condensate. 
This is an explanation from Carl E. Wieman, University of Colorado
