Can we see a spaceship falling into a black hole and entering the event horizon?

Or it pauses in time because the spaceship reaches the speed of light, c?

Answer

There is a powerful time dilation element to what is seen by a distant observer, and the falling object appears to grow every slower (as well as ever more red shifted) as it approaches the event horizon. Asymptotically so, such that (if you are observing on a long enough wavelength) the falling object never quite vanishes.