But what that speed is varies by the properties of local space.
This is the loophole that holds out a possibility of effectively-FTL travel.
To review - gravity warps space-time by increasing its density - the higher the gravity, the more dense space-time becomes. This goes until you reach the edge of a black hole - where spacetime becomes infinitely dense. (Effectively, fluid space-time becomes a solid at the event horizon.)
Outside a black hole, spacetime is measured with real numbers. Inside, you have to use complex numbers, as the gravity increases beyond the limits of the speed of light. Review - with the speed of light, C, the relationship between gravity G and spacetime density D (lower is more dense in this scale) becomes G squared plus D squared = C squared. With units chosen to make the speed of light 1, this simplifies to G^2 + D^2 = 1, or D = Sqrt(1 - G^2). This means that when G>1 (as it does inside an event boundary), D must begin using complex numbers. As G increases to 1.414... the complex value approaches 1 (the density of normal spacetime in the absence of gravity distortions), and then exponentially surpasses it. This means that the density of spacetime, in the presence of sufficient gravity, is less than that of spacetime in the absence of gravity, and this lack of density increases exponentially as the gravity increases. So, light would travel exponentially faster in these regions.
Black holes keep getting bigger on the inside as their gravity increases. The speed of light is constant, but the local spacetime properties change with gravity, to make it effectively faster or slower.
What can we do with this? Traveling across the inside of a black hole towards its center would seem to have very little utility. But, theoretically, if the density of spacetime can take an imaginary component somewhere, it could have an imaginary component anywhere. And that's where we come to effectively-FTL travel.
If some means could be found to make a portion of normal (real number) spacetime exhibit the properties of complex spacetime, in a controllable fashion, we would have discovered an alternate reality similar to Star Trek's® subspace, where the speed of light would be warped to seem faster.
There doesn't seem to be any easy way to do this - after all, if it were easy, we would probably notice it happening somewhere. Unless you count ghostly apparitions and spirits as being able to transit through complex space, this sort of behavior is simply unobserved in everyday life. And I don't know of any way to predict what the physical laws would be like in a universe ruled by complex numbers.
Hopefully, some clever engineer will figure out how to do this someday, and we could then journey to other stars in less than a lifetime.