Technically, no. However there are metamaterials in which certain wavelengths exhibit a negative index of refraction, which cause light to behave in some ways similarly to if it were moving faster.

https://en.wikipedia.org/wiki/Negative-index_metamaterial

Kind of, with trickery.

https://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLig...

Probably not, to get light to move faster, you don't need a new medium - you just need less of the universe getting in its way.