We report the quasi-differential upper limits of extremely high energy (EHE) neutrino flux above 10 PeV based on the analysis of nine years of IceCube data. A complete frequentist approach to calculate the differential limit using the Poisson binned likelihood is developed. It enables the limit to be set in the presence of unknown astrophysical neutrino flux. An event with deposited energy clearly above 1 PeV was detected in addition to two events found in the previous EHE neutrino search. They are consistent with the astrophysical neutrino flux of a power-law-like spectrum but incompatible with predictions of cosmogenic neutrino fluxes with spectrum peaks at energies well above the PeV range. Thus, they are considered as the bulk of background events in setting the limits on EHE neutrino fluxes. The resultant differential upper limit is the most stringent to date in the energy range between $5 \times 10^{6}$ and $5 \times 10^{10}$ GeV. This result indicates that cosmogenic neutrino models that predict a three-flavor neutrino flux of $E_\nu^2\phi_{\nu_e+\nu_\mu+\nu_\tau}\simeq2\times 10^{-8}\ {\rm GeV}/{\rm cm}^2\ \sec\ {\rm sr}$ at $10^9\ {\rm GeV}$ are constrained, bounding a significant parameter space on EHE neutrino models, which assumes a composition of proton-dominated ultra-high-energy cosmic rays.