The answer to your question depends on fine definitions.
Locally the speed of light is always the same; more precisely, the universal, Lorentz invariant speed $c$ (which is also the maximum speed of a cause-effect relationship and experimentally observed to be the same as the speed of light) is constant. This means that any measurement of light speed in any laboratory with small enough time and spatial extent[1] will always yield the same value $c$;
Globally the speed of light can vary. A light clock's period can indeed be observed from afar to vary by an observer at a different point in a gravitational field: witness - for example - the varying $g_{0\,0}$ term in the Schwarzschild or Rindler metrics.
However, the failure of light to escape a black hole is not well thought of in terms of Newtonian notions, although if you calculate the Schwarzshild radius with Newtonian physics you will indeed get the right answer! The energy of the light decreases as described by Newtonian physics, but this manifests itself as a redshift rather than a local "slowing" as discussed in David Hammen's answer. In Newtonian mechanics, the analogous concept to a black hole is called a Dark Star and, in this paradigm, you can escape a Dark Star by climbing a rope dangled from a passing spaceship. You cannot do the same thing from a Schwarzschild black hole without going backwards in time.
[1] Here we mean in the Weierstrass-style limit sense: as we make our laboratory smaller and smaller so that the spacetime manifold appears more and more like a small chunk of flat Minkowskian spacetime, the limiting result in this thought experiment is always $c$.