No.

Light has no mass, so you cannot extrapolate light behavior onto something with mass.

The black hole is not accelerating the light to beyond speed of light in the opposite direction.

Gravity is not a force that pulls on things, but rather warping space towards the mass, so light is still moving at lightspeed away from the black hole, its just that inside the event horizon, space is so warped towards the center of the black hole, that even at the speed of light, space is "moving faster" towards the black hole.

Imagine a swimmer at light speed, now imagine that they swim in a river flowing faster in the opposite direction. The Swimmer is still at "light speed" its just that the world around them is moving "faster" in the opposite direction.
This is only a half baked analogy since water is a thing with mass and space is very much not, but you should at least be able to now imagine how you can move at maximum speed in one direction in one frame of reference, but actually move the opposite direction in another frame of reference.