A search for cosmic neutrino sources using the data collected with the ANTARES detector between early 2007 and the end of 2015 is presented. For the first time, all neutrino interactions are considered in a search for point-like sources, instead of only muon neutrino charged current interactions. This is achieved by using a novel reconstruction algorithm for shower-like events in addition to the standard muon track reconstruction. In order to find the clusters of neutrinos from point-like sources over the randomly distributed atmospheric neutrino events, a maximum likelihood ratio approach is followed. The search for astrophysical neutrino sources is performed with four strategies. In the full-sky search, the whole visible sky of ANTARES is scanned to find the most significant cluster of events. In the second approach, the directions of a pre-defined list of known sources which are potential neutrino emitters are investigated. The third search is similar to the full-sky search but restricted to a region around the Galactic Center (GC). Finally, the fourth approach tests the location of Sagittarius A* as an extended source by assuming a Gaussian emission profile of various widths. No significant evidence of cosmic neutrino sources is found. The most significant cluster in the full-sky search is located at $(\alpha,\delta) = (343.8^{\circ}, 23.5^{\circ})$ with a post-trial significance of $1.9\sigma$. Upper limits on the total neutrino flux from the investigated astrophysical candidates are set between $0.60 \times 10^{-8}$ and $2.1 \times 10^{-8}$ GeV cm$^{2}$ s$^{-1}$. These searches provide the most sensitive limits for a large fraction of the Southern Sky, especially at neutrino energies below 100 TeV.