  We present the first attempt to detect outflows from galaxies approaching the
Epoch of Reionization (EoR) using a sample of 9 star-forming ($\rm SFR=31\pm
20~M_{\odot}~yr^{-1}$) $z\sim 5.5$ galaxies for which the [CII]158$\mu$m line
has been previously obtained with ALMA. We first fit each line with a Gaussian
function and compute the residuals by subtracting the best fitting model from
the data. We combine the residuals of all sample galaxies and find that the
total signal is characterised by a flux excess of $\sim 0.5$ mJy extended over
$\sim 1000$ km~s$^{-1}$. Although we cannot exclude that part of this signal is
due to emission from faint satellite galaxies, we show that the most probable
explanation for the detected flux excess is the presence of broad wings in the
[CII] lines, signatures of starburst-driven outflows. We infer an average
outflow rate of $\rm \dot{M}=54\pm23~ M_{\odot}~yr^{-1}$, providing a loading
factor $\eta=\rm \dot{M}/SFR=1.7\pm1.3$ in agreement with observed local
starbursts. Our interpretation is consistent with outcomes from zoomed
hydro-simulations of {\it Dahlia}, a $z\sim 6$ galaxy ($\rm SFR\sim 100~\rm
M_{\odot}~yr^{-1}$) whose feedback-regulated star formation results into an
outflow rate $\rm \dot{M}\sim 30~ M_{\odot}~yr^{-1}$. The quality of the ALMA
data is not sufficient for a detailed analysis of the [CII] line profile in
individual galaxies. Nevertheless, our results suggest that starburst-driven
outflows are in place in the EoR and provide useful indications for future ALMA
campaigns. Deeper observations of the [CII] line in this sample are required to
better characterise feedback at high-$z$ and to understand the role of outflows
in shaping early galaxy formation.
