We report a quasi-differential upper limit on the extremely-high-energy (EHE) neutrino flux above $5times 10^{6}$ GeV based on an analysis of nine years of IceCube data. The astrophysical neutrino flux measured by IceCube extends to PeV energies, and it is a background flux when searching for an independent signal flux at higher energies, such as the cosmogenic neutrino signal. We have developed a new method to place robust limits on the EHE neutrino flux in the presence of an astrophysical background, whose spectrum has yet to be understood with high precision at PeV energies. A distinct event with a deposited energy above $10^{6}$ GeV was found in the new two-year sample, in addition to the one event previously found in the seven-year EHE neutrino search. These two events represent a neutrino flux that is incompatible with predictions for a cosmogenic neutrino flux and are considered to be an astrophysical background in the current study. The obtained limit is the most stringent to date in the energy range between $5 times 10^{6}$ and $5 times 10^{10}$ GeV. This result constrains neutrino models predicting a three-flavor neutrino flux of $E_ u^2phi_{ u_e+ u_mu+ u_tau}simeq2times 10^{-8} {rm GeV}/{rm cm}^2 sec {rm sr}$ at $10^9 {rm GeV}$. A significant part of the parameter-space for EHE neutrino production scenarios assuming a proton-dominated composition of ultra-high-energy cosmic rays is excluded.
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