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 pointlike 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 unbinned maximum-likelihood searches for PeV gamma-rays in the Southern Hemisphere using 5 yr 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 similar to 0.6 and 100 PeV. Our measurements of pointlike and diffuse Galactic emission of PeV gamma-rays are consistent with the background, so we constrain the angle-integrated diffuse gamma-ray flux from the Galactic plane at 2 PeV to 2.61 x 10(-19) cm(-2) s(-1) TeV-1 at 90% confidence, assuming an E-3 spectrum, and we estimate 90% upper limits on pointlike emission at 2 PeV between 10(-21) and 10(-20) cm(-2) s(-1) TeV-1 for an E-2 spectrum, depending on decl. Furthermore, we exclude unbroken power-law emission up to 2 PeV for several TeV gamma-ray sources observed by the High Energy Spectroscopic System 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 IceCube's high-energy starting event sample. These are currently the strongest constraints on PeV gamma-ray emission.