If one were to drop, say, a pebble, from a sufficient distance, into a black hole. The pebble would continue to accelerate until it were just short of the speed of light. In the next moment it's speed should exceed the speed of light. Why not?
Gravity is no different from any other force. Doesn't matter it comes from a black hole, a star, a planet, or a lump of cheese floating in outer space. It's just a force, like any other.
Put a very powerful rocket engine on the pebble. In a short time it would accelerate the pebble near the speed of light. In the next moment it should exceed the speed of light. Why not?
Because that's not how it works. As you accelerate more and more, time dilation and space contraction also get stronger, so it gets harder and harder to get even closer to the speed of light, no matter what's pushing or pulling the pebble, be that a giant rocket engine or the enormous gravity of a large black hole. Same thing. The faster the pebble goes, the more warped space and time become, the harder it is to increase the speed.
It cannot even reach the speed of light because then time dilation, etc., would become infinite, effectively preventing it from reaching speed of light.
You cannot cheat relativity.
Note: This is not a rigorous explanation but it's close enough and should provide some intuitive understanding of what's going on.