  (Abridged) The HST Frontier Fields project started at the end of 2013 with
the aim of providing extremely deep images of 6 massive galaxy clusters. One of
the main goals of this program is to push several telescopes to their limits in
order to provide the best current view of the earliest stages of the Universe.
We present a detailed analysis of $z$$\sim$8 objects behind the HFFs lensing
cluster, MACS0416-2403, combining 0.3-1.6 $\mu$m imaging from HST, ground-based
$K_s$ imaging from VLT HAWK-I, and 3.6 $\mu$m and 4.5 $\mu$m Spitzer Space
Telescope. We apply the classical Lyman Break technique, combining
non-detection criteria and color-selection. In order to avoid contamination by
mid-$z$ interlopers, we require a strong break between optical and near
infrared data. The luminosity function at $z$$\sim$8 is computed using a MC
method taking advantage of the SED-fitting results. A piece of cautionary
information is gleaned from new deep optical photometry of a previously
identified $z$$\sim$8 galaxy in this cluster, which is now firmly detected as a
mid-$z$ interloper with a strong $\approx1.5$ mag Balmer break (between F606W
and F125W). Using the SED of this interloper, we estimate the contamination
rate of our MACS0416$-$2403 sample, as well as for previous samples in Abell
2744 based on HFF data. Our selection recovers 4 robust objects with
m$_{F160W}$ ranging from 26.0 to 27.9 AB and located in modest amplification
regions ($\mu<$2.4). Two of the objects display a secondary break between the
IRAC 3.6 $\mu$m and 4.5 $\mu$m bands which could be associated to the Balmer
break or emission lines at $z$$\sim$8. The candidates generally have star
formation rates around $\sim$10 M$_{\odot}$/yr and sizes ranging from 0.2 to
0.5 kpc, in good agreement with previous observations and expectations for
objects in the early Universe.
