We employ radio-frequency spectroscopy to investigate a polarized spin mixture of ultracold Li-6 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.
