You have to be careful in how you define speed before answering this question: do you mean group velocity or phase velocity? And faster than the local velocity

You have to be careful in how you define speed before answering this question: do you mean group velocity or phase velocity? And faster than the local velocity of light or the value in vacuo? It is indeed possible for a particle to travel at a speed greater than the local phase velocity of light through a medium. This is what brings about Čerenkov radiation, which is that eerie blue glow associated with the core of nuclear reactors.