We use the surface detector of the Pierre Auger Observatory to search for air showers initiated by photons with an energy above 1019 eV. Photons in the zenith angle range from 30∘ to 60∘ can be identified in the overwhelming background of showers initiated by charged cosmic rays through the broader time structure of the signals induced in the water-Cherenkov detectors of the array and the steeper lateral distribution of shower particles reaching ground. Applying the search method to data collected between January 2004 and June 2020, upper limits at 95% CL are set to an E-2 diffuse flux of ultra-high energy photons above 1019 eV, 2 × 1019 eV and 4 × 1019 eV amounting to 2.11 × 10-3, 3.12 × 10-4 and 1.72 × 10-4 km-2 sr-1 yr-1, respectively. While the sensitivity of the present search around 2 × 1019 eV approaches expectations of cosmogenic photon fluxes in the case of a pure-proton composition, it is one order of magnitude above those from more realistic mixed-composition models. The inferred limits have also implications for the search of super-heavy dark matter that are discussed and illustrated.