  We investigate the properties of a strongly interacting imbalanced mixture of
bosonic $^{41}$K impurities immersed in a Fermi sea of ultracold $^6$Li atoms.
This enables us to explore the Fermi polaron for large impurity concentrations
including the case where they form a Bose-Einstein condensate. The system is
characterized by means of radio-frequency injection spectroscopy for tunable
interactions using an interspecies Feshbach resonance. We find that the energy
of the Fermi polarons formed in the thermal fraction of the impurity cloud
remains rather insensitive to the impurity concentration, even as we approach
equal densities for both species. The apparent insensitivity to high
concentration is consistent with the theoretical prediction, based on Landau's
quasiparticle theory, of a weak effective interaction between the polarons. The
condensed fraction of the bosonic $^{41}$K gas is much denser than its thermal
component, which leads to a break-down of the Fermi polaron description.
Instead, we observe a new branch in the radio-frequency spectrum with a small
energy shift, which is consistent with the presence of Bose polarons formed by
$^{6}$Li fermions inside the $^{41}$K condensate. A closer investigation of the
behavior of the condensate by means of Rabi oscillation measurements support
this observation, indicating that we have realized Fermi and Bose polarons, two
fundamentally different quasiparticles, in one cloud.
