  The PROSPECT experiment is designed to perform precise searches for
antineutrino disappearance at short distances (7 - 9~m) from compact nuclear
reactor cores. This Letter reports results from a new neutrino oscillation
analysis performed using the complete data sample from the PROSPECT-I detector
operated at the High Flux Isotope Reactor in 2018. The analysis uses a
multi-period selection of inverse beta decay neutrino interactions with reduced
backgrounds and enhanced statistical power to set limits on electron-flavor
disappearance caused by mixing with sterile neutrinos with 0.2 - 20 eV$^2$ mass
splittings. Inverse beta decay positron energy spectra from six different
reactor-detector distance ranges are found to be statistically consistent with
one another, as would be expected in the absence of sterile neutrino
oscillations. The data excludes at 95% confidence level the existence of
sterile neutrinos in regions above 3~eV$^2$ previously unexplored by
terrestrial experiments, including all space below 10~eV$^2$ suggested by the
recently strengthened Gallium Anomaly. The best-fit point of the Neutrino-4
reactor experiment's claimed observation of short-baseline oscillation is ruled
out at more than five standard deviations.
