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.