A search for cosmic neutrino sources using the data collected with the ANTARES neutrino telescope between early 2007 and the end of 2015 is performed. For the first time, all neutrino interactions—charged- and neutral-current interactions of all flavors—are considered in a search for point-like sources with the ANTARES detector. In previous analyses, only muon neutrino charged-current interactions were used. This is achieved by using a novel reconstruction algorithm for shower-like events in addition to the standard muon track reconstruction. The shower channel contributes about 23% of all signal events for an E-2 energy spectrum. No significant excess over background is found. The most signal-like cluster of events is located at (a,d)=(343.8°,23.5°) with a significance of 1.9s. The neutrino flux sensitivity of the search is about E2dF/dE=6×10-9¿¿GeV¿cm-2¿s-1 for declinations from -9° up to -42°, and below 10-8¿¿GeV¿cm-2¿s-1 for declinations up to 5°. The directions of 106 source candidates and 13 muon track events from the IceCube high-energy sample events are investigated for a possible neutrino signal and upper limits on the signal flux are determined.