  Gravitational collapse is still poorly understood in the context of $f(R)$
theories of gravity, since the Oppenheimer-Snyder model is incompatible with
their junction conditions. In this work, we will present a systematic approach
to the problem. Starting with a thorough analysis of how the Oppenheimer-Snyder
construction should be generalised to fit within metric $f(R)$ gravity, we
shall subsequently proceed to explore the existence of novel exterior solutions
compatible with physically viable interiors. Our formalism has allowed us to
show that some paradigmatic vacuum metrics cannot represent spacetime outside a
collapsing dust star in metric $f(R)$ gravity. Moreover, using the junction
conditions, we have found a novel vacuole solution of a large class of $f(R)$
models, whose exterior spacetime is documented here for the first time in the
literature as well. Finally, we also report the previously unnoticed fact that
the Oppenheimer-Snyder model of gravitational collapse is incompatible with the
Palatini formulation of $f(R)$ gravity.
