The measurement of diffuse PeV gamma-ray emission from the Galactic plane would provide information about the energy spectrum and propagation of Galactic cosmic rays, and the detection of a point-like source of PeV gamma rays would be strong evidence for a Galactic source capable of accelerating cosmic rays up to at least a few PeV. This paper presents several un-binned maximum likelihood searches for PeV gamma rays in the Southern Hemisphere using 5 years of data from the IceTop air shower surface detector and the in-ice array of the IceCube Observatory. The combination of both detectors takes advantage of the low muon content and deep shower maximum of gamma-ray air showers, and provides excellent sensitivity to gamma rays between $sim$0.6 PeV and 100 PeV. Our measurements of point-like and diffuse Galactic emission of PeV gamma rays are consistent with background, so we constrain the angle-integrated diffuse gamma-ray flux from the Galactic Plane at 2 PeV to $2.61 times 10^{-19}$ cm$^{-2}$ s$^{-1}$ TeV$^{-1}$ at 90% confidence, assuming an E$^{-3}$ spectrum, and we estimate 90% upper limits on point-like emission at 2 PeV between 10$^{-21}$ - 10$^{-20}$ cm$^{-2}$ s$^{-1}$ TeV$^{-1}$ for an E$^{-2}$ spectrum, depending on declination. Furthermore, we exclude unbroken power-law emission up to 2 PeV for several TeV gamma-ray sources observed by H.E.S.S., and calculate upper limits on the energy cutoffs of these sources at 90% confidence. We also find no PeV gamma rays correlated with neutrinos from IceCubes high-energy starting event sample. These are currently the strongest constraints on PeV gamma-ray emission.
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