Radio-frequency spectroscopy is used to identify a long-lived, metastable quasiparticle in a lithium–potassium Fermi mixture with strongly repulsive interactions. Metastable states in Fermi gases with strong repulsive interactions are of fundamental interest, but the realization of such systems is challenging because they are intrinsically unstable against decay. Two groups have overcome this obstacle and report the detection of the theoretically predicted repulsive Fermi polarons. Kohstall et al. study a three-dimensional system of ultracold potassium impurities resonantly interacting with a Fermi sea of lithium atoms. The character of the interaction stabilizes the repulsive regime, enabling the authors to detect long-lived, metastable repulsive polarons. Koschorreck et al. study both attractive and repulsive Fermi polarons in a two-dimensional, spin-imbalanced Fermi gas of potassium atoms, and find evidence for a pairing transition. The results from these two studies hold promise for the creation of exotic states with ultracold fermionic atoms, such as ferromagnetic quantum phases. Ultracold Fermi gases with tunable interactions provide a test bed for exploring the many-body physics of strongly interacting quantum systems1,2,3,4. Over the past decade, experiments have investigated many intriguing phenomena, and precise measurements of ground-state properties have provided benchmarks for the development of theoretical descriptions. Metastable states in Fermi gases with strong repulsive interactions5,6,7,8,9,10,11 represent an exciting area of development. The realization of such systems is challenging, because a strong repulsive interaction in an atomic quantum gas implies the existence of a weakly bound molecular state, which makes the system intrinsically unstable against decay. Here we use radio-frequency spectroscopy to measure the complete excitation spectrum of fermionic 40K impurities resonantly interacting with a Fermi sea of 6Li atoms. In particular, we show that a well-defined quasiparticle exists for strongly repulsive interactions. We measure the energy and the lifetime of this 'repulsive polaron'9,12,13, and probe its coherence properties by measuring the quasiparticle residue. The results are well described by a theoretical approach that takes into account the finite effective range of the interaction in our system. We find that when the effective range is of the order of the interparticle spacing, there is a substantial increase in the lifetime of the quasiparticles. The existence of such a long-lived, metastable many-body state offers intriguing prospects for the creation of exotic quantum phases in ultracold, repulsively interacting Fermi gases.