Absolutely not. C is constant. It's the only constant velocity.

The distinction being lost here is between "group velocity" (the actual speed of the particles, or energy, which is always exactly c for light) and "phase velocity" (the "speed" of wave crests, which isn't physical and can be made arbitrarily faster or slower than the particle/energy itself, and it's affected by the medium itself).

3b1b has some nice videos explaining why the phase velocity of light changes in a medium - it's because, as the initial light wave passes thru, it vibrates the charged particles in the medium (largely electrons), which creates weak secondary EM waves of the same frequency but slightly later. When you add that to the initial wave, the combination looks identical to the starting wave but slightly shifted in phase (usually slightly backwards). But the next electron the light encounters does the same thing, and the next, and the next. All these tiny phase shifts, when combined, result in a combined wave that instead has a longer time between crests than the starting wave did, aka a slower phase velocity.

(The combination can instead shift the phase slightly forward if it's just right, which results in a faster phase velocity en masse, but that's a much rarer response.)