My understanding of it is that massive objects do not affect the speed of light at all, not even black holes.

They bend space itself meaning that light will always travel at the same speed, even in proximity of a black hole, but over a distance that becomes, from the local point of view, more and more vast, to the point where, seen from the "undeformed" space outside, it looks as if it took it longer to traverse that stretch of space (because we cannot perceive and have a hard time even imagining the fact that "there is more space" there), or it doesn't come out of it at all.

Maybe someone more into these aspects can tell me if I am oversimplifying or outright wrong.