With the Auger Engineering Radio Array (AERA) of the Pierre Auger Observatory, we have observed the radio emission from 561 extensive air showers with zenith angles between 60$^\circ$ and 84$^\circ$. In contrast to air showers with more vertical incidence, these inclined air showers illuminate large ground areas of several km$^2$ with radio signals detectable in the 30 to 80\,MHz band. A comparison of the measured radio-signal amplitudes with Monte Carlo simulations of a subset of 50 events for which we reconstruct the energy using the Auger surface detector shows agreement within the uncertainties of the current analysis. As expected for forward-beamed radio emission undergoing no significant absorption or scattering in the atmosphere, the area illuminated by radio signals grows with the zenith angle of the air shower. Inclined air showers with EeV energies are thus measurable with sparse radio-antenna arrays with grid sizes of a km or more. This is particularly attractive as radio detection provides direct access to the energy in the electromagnetic cascade of an air shower, which in case of inclined air showers is not accessible by arrays of particle detectors on the ground.