  The IceCube Collaboration has observed a high-energy astrophysical neutrino
flux and recently found evidence for neutrino emission from the blazar TXS
0506+056. These results open a new window into the high-energy universe.
However, the source or sources of most of the observed flux of astrophysical
neutrinos remains uncertain. Here, a search for steady point-like neutrino
sources is performed using an unbinned likelihood analysis. The method searches
for a spatial accumulation of muon-neutrino events using the very
high-statistics sample of about $497\,000$ neutrinos recorded by IceCube
between 2009 and 2017. The median angular resolution is $\sim1^\circ$ at 1 TeV
and improves to $\sim0.3^\circ$ for neutrinos with an energy of 1 PeV. Compared
to previous analyses, this search is optimized for point-like neutrino emission
with the same flux-characteristics as the observed astrophysical muon-neutrino
flux and introduces an improved event-reconstruction and parametrization of the
background. The result is an improvement in sensitivity to the muon-neutrino
flux compared to the previous analysis of $\sim35\%$ assuming an $E^{-2}$
spectrum. The sensitivity on the muon-neutrino flux is at a level of $E^2
\mathrm{d} N /\mathrm{d} E = 3\cdot
10^{-13}\,\mathrm{TeV}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. No new evidence for
neutrino sources is found in a full sky scan and in an a priori candidate
source list that is motivated by gamma-ray observations. Furthermore, no
significant excesses above background are found from populations of
sub-threshold sources. The implications of the non-observation for potential
source classes are discussed.
