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.