  We investigate the zero-temperature quasiparticle properties of a mobile
impurity immersed in a strongly interacting Fermi superfluid at the crossover
from a Bose-Einstein condensate (BEC) to a Bardeen--Cooper--Schrieffer (BCS)
superfluid, by using a many-body $T$-matrix approach that excludes Efimov
trimer bound states. Termed BEC-BCS crossover polaron, or crossover polaron in
short, this quasiparticle couples to elementary excitations of a many-body
background and therefore could provide a useful probe of the underlying
strongly interacting Fermi superfluid. Due to the existence of a significant
pairing gap $\Delta$, we find that the repulsive polaron branch becomes less
well-defined. In contrast, the attractive polaron branch is protected by the
pairing gap and becomes more robust at finite momentum. It remains as a
delta-function peak in the impurity spectral function below a threshold
$2\Delta$. Above the threshold, the attractive polaron enters the particle-hole
continuum and starts to get damped. We predict the polaron energy, residue and
effective mass for realistic Bose-Fermi mixtures, where the minority bosonic
atoms play the role of impurity. These results are practically useful for
future cold-atom experiments on crossover polarons.
