@BartoszMilewski - trust the equivalence principle: any small enough patch of spacetime is indistinguishable from Minkowski spacetime for a free-falling particle.
If you fall through the event horizon of an enormous black hole with your arm outstretched before you, your hand doesn't disappear as it crosses the horizon. But if you use rockets to hover outside the horizon and stick your arm in, it gets ripped off and disappears from view.