We present the interpretation of the muon and scintillation signals of ultrahigh-energy air showers observed by AGASA and Yakutsk extensive air shower array experiments. We consider case-by-case ten highest-energy events with known muon content and conclude that at the 95% confidence level none of them was induced by a primary photon. Taking into account statistical fluctuations and differences in the energy estimation of proton and photon primaries, we derive an upper limit of 36% at a 95% confidence level on the fraction of primary photons in the cosmic-ray flux above 10{sup 20} eV. This result disfavors the Z-burst and superheavy dark-matter solutions to the Greisen-Zatsepin-Kuzmin-cutoff problem.