I know a black hole can absorb matter and maybe also release Hawking radiation. I think that according to the Gullstrand–Painlevé coordinates, space is flowing and the apparent horizon is the region of space moving outward at the speed of light that will never fall into the black hole or escape. I know what matter the black hole absorbs later affects where the apparent horizon is not according to that coordinate system. However, maybe slightly different physical laws could end up with the following result. All the space outside the apparent horizon is skewed forward in time compared to the Gullstrand–Painlevé coordinate system and the amount it's skewed forward varies as the log of the reciprocal of the distance from the apparent horizon. I don't see how it's impossible that the laws would make it actually for real take an infinite amount of time for an object falling into a black hole to reach the apparent horizon. Maybe that would be a solution to the black hole information paradox. Maybe there's some sort of real actual physical truth out there that's true whether we define it to be or not we just don't know it and the truth of what's really going on means you cannot reasonably define it to take a finite amount of time to reach the apparent horizon, and the space beyond it doesn't exist whether you define it to or not.
The matter falling into a black hole does reach the event horizon, so it is not possible. However, a better question would be that if a black hole has infinite density, than does matter actually reach the singularity? The answer here has to be yes, mainly because if not, then nothing we know of could undo a black hole, but also because while it has infinite density, than it also warps space-time infinitely, at least at the singularity. So at some point, the matter should merge into the singularity's space-time warp and by then it is practically a part of it.
So the answer to your original question would be a no, but that is according to our present theories. It is an intriguing question, however, and it doesn't hurt to speculate.