Information can’t travel faster than light, things can.
Researchers of Australian Attosecond Science Facility (AASF) did exactly that. They found that this simplest of transitions takes at most 1.8 attoseconds (1.8 * 10 -18s).
That means, at the speed of light, we are talking about travelling the distance of just 5.4 angstroms (.54 nanometre).
While this has fascinating implications for the practical apllications of, say, the construction of a quantum limited transistor, “instantaneous” in this context doesn’t mean that it violates Einstein’s Relativity.
It isn’t as though at one instant you can say “the particle is over there”, and then, some tiny amount of time later you can say, “the particle is now located here instead”, with that change in distance divided by the change in time exceeding the speed of light.
The experiment, which is remarkable for how precise and clean it was in only involving a single particle in a single, bound system, simply shows that there’s no fundamental quantum delay in this tunneling transition.