The only information I could find about something being infinitely ductile is this answer which pretty much states that gold cannot undergo ductile fracture at all. I define a material to be infinitely ductile if it can't support crack propagation at all and can keep being stretched thinner without breaking and define one infinitely ductile material to be stronger than another infinitely ductile material if it requires tension stress in order to undergo permanent deformation of being stretched thinner. An infinitely ductile material could be so useful as long as it's not so weak like gold. It could support so much weight without fear that it will undergo fatigue and collapse. If you want two spheres made of an infinitely ductile material not to deform on collision, they'll have to be made of a stronger material than gold because part of them is subject to a lot of stress on collision and gold is so weak. Maybe for making thin wires, gold is good enough because all you have to do is keep repeating the action that doubles its length. That answer also seems to suggest that only a material that doesn't react with air to form a passivating layer can be infinitely ductile. I don't see why that must be the case.
Also, the strongest infinitely ductile material might also have the unfortunate property of being an unstable substance that slowly reverts to another substance or a heterogeneous mixture of two substances that is not infinitely ductile. If that's the case, I also want to know what's the strongest know infinitely ductile material that has the additional property of being a stable substance. It doesn't matter that it not slowly react with the air around it. It just matters that it not self react and revert to another material.