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 $E^2 mathrm{d} N /mathrm{d} E = 3cdot 10^{-13},mathrm{TeV},mathrm{cm}^{-2},mathrm{s}^{-1}$. 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.
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