Information can’t travel faster than light, things can.
Whether you are talking about a particle at rest, a free particle traveling through space, a bound particle (like an electron in an atom) that’s restricted as far as where it’s allowed to be, or a particle that encounters an obstacle that restricts what quantum states it’s allowed to occupy; there are no certainties until you make a measurement. Only probabilities!
So you might think, if you have a system that has the probability of tunneling from one side of the quantum barrier (like bound in an atom, or in a false minimum) to the other, there would be a restriction on how quickly that transition could occur.
Maybe it would depend on the size of the barrier, the thickness of the barrier, or some other factor that was related to its physical properties. After all, in this universe, everything should be limited by the speed of light.
The simplest setup of all is to take one single particle, like an electron, bound in a restricted system, like a hydrogen atom.
There’s a finite non-zero probability that it will tunnel to an unbound state.
By imaging it with proper equipment, ultra-fast photons, for instance, you can accurately measure the amount of time it takes to tunnel from a bound to an unbound state.