I have several ideas on this.
Materials exist in the Cold Regime
Things we consider solid at our normal temperatures lose any elacticity and ductility before getting down to 4K.
But what about materials that could not exist at our temperature? As we found when exploring silicon-based life, direct analogs of organic molecules using Si for C would require cryogenic temperatures to exist. So it might be possible to design something that has enough bonding strength to hold together but not too much as to prevent all movement, at 4K. At significantly warmer temperatures the molecules would fly apart.
This could be true for both elastic (rubbery) and ductile (metal) materials.
Material Withstands Being Brittle
A material may be too brittle, but it can still be very hard. If there are no flaws in the microstructure, it might prove very difficult to chip at all, and any chip you do make will be small spallations (the force only goes down as it spreads out). So a perfect crystal may be just fine in terms of strength.
There are also composite materials. If we can’t combine strong and flexible because there is nothing flexible, you at least still have a combination. This will still provide the effect that cracks can’t propigate farther than the breaking of one fiber. The material boundary will interfere with breaking forces, and even reflect the force.
Avoiding the Brittle Regime?
The linked Physics post discusses the ductile-to-brittle transition
Temperature sort of maps to time and information transfer. At high temperatures, particles/dislocations travel quicker and with more ease than at lower temperatures. Thus information (stress, strain, ...) travels through the sample. There is more time to move around and shift to try to alleviate the applied stress or strain.
So... do you know why diamond is a better thermal conductor than copper, even though it’s not a metal? Phonons. If information about the stresses could be carried off by electron density waves which amplify the actual atom displacements of the material, you can avoid super-brittle behavior.
Maybe that’s not true—but it’s an awsome handwave for a science-based not-dumb story!
Ever hear of nitinol? A number of years ago, superelastic nitinol was all the rage for eyeglasses frames and watch bands.
How does solid metal seem to be rubbery? The stress causes a high pressure which causes a phase transition to a smaller crystal. When released, it pops back to the larger form!
So, the atoms don’t rip free of their bond positions and thus it is not damaged. Engineering this property into a material at 4K might be possible, not with a simple alloy, but with a complex material or even grains that act as a meta-material.
Remember this Answer?
Such low temperatures enable super effects like superconductivity, so maybe the solution is to take advantage of that. In my “plausible supermaterial”, tiny bits the size of mineral grains are held in place via flux pinning, overcoming the normal physical strengths of atomic bonds and making overextension reversable rather than damage.
The 4K temperature makes this easier to acheive, with today’s knowledge. You can get ductile behavior on the scale of individual grains, substituting the flux pinning for normal atomic bonds. Even if it doesn’t have the futuristic ability to fly the units back where they belong, a simple mass of this material will exhibit the ductile behavior of a room-temperature metal, with no accumulation of fracture growth or “work hardening”, even as the individual grains are very hard and brittle.
And that’s the answer I had in mind when I saw your initial comment on the subject: engineer the bulk properties using grain-sized units of normal matter, and superconductive effects between them to bypass the limitations in available atomic bonds.