Any family of plastics / filaments that bend and keep their shape? Not asking for a product recommendation, unless there's only one product out there like this...

I have a need for a flexible filament that retains its deformation, kind of like silly putty. Are there any filaments out there that, after you bend or stretch them, they keep their shape?

I'm sure to some degree flexible filaments do this -- Which filament might be best? Commenters -- what's the scientific term for this? 

Actually, metals exhibit this type of flexibility. Do any of the 3D printable metal filaments flex, bend, and keep their shape?
To have a material be "flexible" and retain the shape after the "flexing", you need to apply a stress in excess of the elastic deformation stress.  You will want a material that has the elastic deformation limit that is much lower than the ultimate tensile strength.  Unfortunately, for plastics, this is difficult to find.

Most plastics are made up of tightly bound, long-chain molecules.  Flexing beyond the elastic limit requires breaking these bonds, which introduces weak points in the plastic.  You can see this by bending almost any plastic.  Either it will snap in two, or you will see a light or white colored line along the fold.  The lighter color comes from light scattering from the broken bonds.

"Flexible" structural plastics such as ABS are a copolymer (more than one type of molecule) in which one of the polymers is rubbery.  The rubbery bits provide places within the bulk material where the stress can create strain that doesn't require breaking bonds.

I doubt that any 3D-printer FDM compatible filaments will satisfy your needs.  A filament must keep its shape during printing, if for no other reason than to permit the extruder drive mechanism to apply pressure to the filament.  There are 3D-printable filaments, but they are very elastic and return to their shape when the stress is released.

Some metals can respond to stress with by flowing rather than by elastic strain.  Lead comes to mind.  Someone suggested copper.  There are several degrees of hardness of copper depending on the alloy and the annealing schedule after it was formed.  Copper also work-hardens, becoming harder and more likely break under stress the more the copper flows under stress.