Not really, at a certain amount of current (how much exactly depends of the shape and material of the superconductor in question) it "quenches", where it abruptly goes back to being a material with non-zero resistance. Since there is usually a ton of current already flowing through the material, this usually leads to damage as the former superconductor heats up rapidly.

See also https://en.wikipedia.org/wiki/Superconducting_magnet#Magnet_...

There is a critical magnetic field at which superconductors stop being superconductive [1]. Generally, a superconductor carrying a lot of current suddenly becoming conductive is a bad thing. This is directly related to the maximum current density in a particular superconducting wire, so you need larger cross sections to increase the current carrying capability.

[1] https://en.wikipedia.org/wiki/Critical_field

Nah, but easy misconception. Rearrange it as V=IR and that formula tells you there’s no voltage drop across the superconducting bit regardless of current.

More current produces a stronger magnetic field around the conductor, and superconductors usually have a maximum magnetic field beyond which they stop superconducting.