Can the force of gravity in a black hole, accelerate light past the speed of light?

Gravity slows light down.
I see what you're getting at, we can make light come to a standstill with a Bose-Einstein condensate, and bring it to speeds of about 56kmph.
I would guess that the light could bend around and maybe, and maybe it could into the black hole where it is absorbed into the thickets part, where I would say couldn't go any further because it's supposedly infinitely dense.

Light is relative, if your car was travelling at the speed of light, and you turn your headlights on, the light would go twice the speed of light from the view point of an observer,
Think of the milkyway, and the other galaxies going faster, but away from us, that light will still reach us, but be going slower, and I think because of this a lower frequency and red (red shift, maybe blue I forget which way it works...)

No, the speed of light is always constant. A gravitational field can affect light by changing its
frequency (which is equivalent to a red shift or blue shift), but not its speed. The General Theory
of Relativity describes how light and gravity interact. This particular question has been verified
experimentally in a well-known (to physicists, anyway) experiment known as the Pound-Rebka
Experiment..

Gravity has an influence on the propagation of light. But the influence is almost negligible.
A black hole's gravity does change the velocity of light. It doesnt accelerate light, it retards it. Light tends to bend near a black hole. So it makes an angle say, ? with its original direction of propagation. By the action of the blackhole's gravity, the velocity of light becomes c cos?  (assuming the original direction of light to be horizontal)  which is less than c. So, blackholes slow down light.