The ANTARES detector is at present the most sensitive neutrino telescope in the northern hemisphere. The highly significant cosmic neutrino excess observed by the Antarctic IceCube detector can be studied with ANTARES, exploiting its complementing field of view, exposure, and lower energy threshold. Searches for an all-flavor diffuse neutrino signal, covering nine years of ANTARES data taking, are presented in this Letter. Upward-going events are used to reduce the atmospheric muon background. This work includes for the first time in ANTARES both track-like (mainly ${\nu }_{\mu })$ and shower-like (mainly ${\nu }_{e}$) events in this kind of analysis. Track-like events allow for an increase of the effective volume of the detector thanks to the long path traveled by muons in rock and/or sea water. Shower-like events are well reconstructed only when the neutrino interaction vertex is close to, or inside, the instrumented volume. A mild excess of high-energy events over the expected background is observed in nine years of ANTARES data in both samples. The best fit for a single power-law cosmic neutrino spectrum, in terms of per-flavor flux at 100 TeV, is ${{\rm{\Phi }}}_{0}^{1f}(100\ {\rm{TeV}})=(1.7\pm 1.0)\times $ 10-18 GeV-1 cm-2 s-1 sr-1 with spectral index ${\rm{\Gamma }}={2.4}_{-0.4}^{+0.5}$. The null cosmic flux assumption is rejected with a significance of 1.6s.