  We employ radio-frequency spectroscopy to investigate a polarized
spin-mixture of ultracold ${}^6$Li atoms close to a broad Feshbach scattering
resonance. Focusing on the regime of strong repulsive interactions, we observe
well-defined coherent quasiparticles even for unitarity-limited interactions.
We characterize the many-body system by extracting the key properties of
repulsive Fermi polarons: the energy $E_+$, the effective mass $m^*$, the
residue $Z$ and the decay rate $\Gamma$. Above a critical interaction, $E_+$ is
found to exceed the Fermi energy of the bath while $m^*$ diverges and even
turns negative, thereby indicating that the repulsive Fermi liquid state
becomes energetically and thermodynamically unstable.
