  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
IceCube's high-energy starting event sample. These are currently the strongest
constraints on PeV gamma-ray emission.
