  Neutrinos with energies above $10^{17}$ eV are detectable with the Surface
Detector Array of the Pierre Auger Observatory. The identification is
efficiently performed for neutrinos of all flavors interacting in the
atmosphere at large zenith angles, as well as for Earth-skimming $\tau$
neutrinos with nearly tangential trajectories relative to the earth. No
neutrino candidates were found in $\sim\,14.7$ years of data taken up to 31
August 2018. This leads to restrictive upper bounds on their flux. The $90\%$
C.L. single-flavor limit to the diffuse flux of ultra-high-energy neutrinos
with an $E_\nu^{-2}$ spectrum in the energy range $1.0 \times 10^{17}~{\rm eV}
- 2.5 \times 10^{19}~{\rm eV}$ is $E^2 {\rm d}N_\nu/{\rm d}E_\nu < 4.4 \times
10^{-9}~{\rm GeV~cm^{-2}~s^{-1}~sr^{-1}}$, placing strong constraints on
several models of neutrino production at EeV energies and on the properties of
the sources of ultra-high-energy cosmic rays.
