  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 9 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 9
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
$\Phi_0^{1f}(100\ \textrm{TeV}) = \left(1.7\pm 1.0\right)
\times$10$^{-18}$\,GeV$^{-1}$\,cm$^{-2}$\,s$^{-1}$\,sr$^{-1}$ with spectral
index $\Gamma = 2.4^{+0.5}_{-0.4}$. The null cosmic flux assumption is rejected
with a significance of 1.6$\sigma$.
