No, if you fall into a black hole, it happens in a finite amount of time to you.
It is the observer at infinity that never sees you fall into the black hole, but real physics is local, you have to use the coordinate system of the person falling into the black hole to determine what happens to them.
> It is the observer at infinity that never sees you fall into the black hole
We don't even need an observer to be at infinity, thanks to the expansion of the universe. With some future telescope our descendants may observe something on a trajectory to enter a black hole in an early-universe galaxy that is just crossing that observer's (cosmological) horizon.
I think it's relevant to raise this since the article at the top is about embedding black-hole-like collapsed stars in an expanding universe and the research which directly discusses the observable consequences.
> real physics is local
Yes, absolutely. You still get spaghettified if you fly into a black hole which is the only other appreciable mass left in the far far future of our universe. Nobody needs to see your last moments.
> you have to use the coordinate system of the person falling into the black hole to determine what happens to them
No, you can use any coordinates you want (or no coordinates at all), but you have to be aware that there are quantities which are invariant under changes of coordinates (e.g. the curvature scalars) and quantities which are coordinate-dependent, and that some systems of coordinates make the latter difficult or even impossible to calculate.
Indeed the infaller can use any set of coordinates she or he wants. Some time coordinate (wristwatch? distant pulsars?) and spatial spherical coordinates with the infaller always at the spatial orgin, East-North-Up coordinates originating on the (spinning) black hole, etc. are all (pardon the pun) attractive in these circumstances.
Also, defining exactly where "falling in" happens is tricky, even for the infaller. Visser 2014 on horizons: <https://arxiv.org/abs/1407.7295>, second sentence third paragraph of the Introduction section ("These distinctions even make a difference when precisely defining what a "black hole" is -- the usual definition in terms of an event horizon is mathematically clean, leading to many lovely theorems [20], but bears little to no resemblance to anything a physicist could actually measure.")
The crossing is not at a straightforward conception of "the end of time" in an expanding universe, since most possible observers are carried away from the final fall-in by the expansion of the universe, so there's nobody orbiting "at infinity" who could in principle see the infall take "an infinite time".
Horizons are part of the causal structure of the entire universe, black holes, planets, toads, warts, and all. The horizon is dominated by the central mass and spin, but not fully determined by it. The horizon in a close black hole binary (or triple) gets very complicated. ("The horizon" is not even necessarily physically measurable, and with black hole evaporation might not even exist, although there are other features which can be indicative of the point of no return for an infaller).
Preliminaries done, there is the "no drama" conjecture. Given a large enough black hole in a quiet enough setting a freely-falling infaller will not know she or he has passed the point of no return, perhaps for several minutes according to his or her wristwatch.
That's because the tidal curvature at the point of no return gets very small as we take the mass of a slowly-spinning black hole above millions of stellar masses, and that's the curvature that's relevant in spaghettification, the leading cause of death of astronauts entering isolated black holes.
Of course, most of the black holes we have found are far from isolated (otherwise we probably wouldn't see them with current equipment), so an infaller is likely to be blasted apart by hard X-rays and superhot gas instead of falling straight in.
The observables for something strongly accelerating into a black hole for a faraway orbiting observer can be quite different; unlike for speed there is no maximum acceleration in relativity. One would have to find a limit to acceleration in the behaviour of matter. An astronaut is not going to survive anything like the acceleration needed to make much difference to the distant orbiting observer though.
The distant observer in the not-really-our-universe Schwarzschild model and seeing the infinitely-prolonged final infall is at rest with respect to the central mass. Different observers, e.g. ones shooting themselves into the same black hole, or hovering just above a different black hole, can see qualitatively different things.
Generically, outside observers will see a dimming and shrinking of (practically) any infaller closer to the black hole than the observer. Many such observers will lose sight of the infaller before the infaller has truly hit a point of no return. Consequently some observers could find themselves seeing a presumed-lost astronaut grow brighter and bigger again, and leave the vincinity of the black hole. (Substitute gas, dust, and parts of stars for astronaut in the previous sentence, and that is what the Event Horizon Telescope collaboration, among others, searches for.)
It is the observer at infinity that never sees you fall into the black hole, but real physics is local, you have to use the coordinate system of the person falling into the black hole to determine what happens to them.