  We use the surface detector of the Pierre Auger Observatory to search for air
showers initiated by photons with an energy above $10^{19}$ eV. Photons in the
zenith angle range from 30$^\circ$ to 60$^\circ$ 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 $10^{19}$ eV, $2{\times}10^{19}$ eV and
$4{\times}10^{19}$ eV amounting to $2.11{\times}10^{-3}$, $3.12{\times}10^{-4}$
and $1.72{\times}10^{-4}$ km$^{-2}$ sr$^{-1}$ yr$^{-1}$, respectively. While
the sensitivity of the present search around $2 \times 10^{19}$ 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.
