  The evolution of dwarf galaxies is dramatically affected by gaseous and dusty
outflows, which can easily deprive their interstellar medium of the material
needed for the formation of new stars, simultaneously enriching their
surrounding circumgalactic medium (CGM). In this letter, we present the first
evidence of extended [CII] 158 $\mu$m line and dust continuum emission in local
dwarf galaxies hosting star-formation-driven outflows. By stacking the [CII],
far-infrared, and near-UV (NUV) emission obtained from Herschel and GALEX data,
we derived the average radial profiles, and compared the spatial extension of
gas, dust, and stellar activity in dwarf galaxies. We find that [CII] and dust
emissions are comparable to each other, and more extended than the NUV
continuum. The [CII] size is in agreement with that measured for $z>4$
star-forming galaxies, suggesting that similar mechanisms could be at the
origin of the observed atomic carbon reservoir around local and high-$z$
sources. The cold dust follows the [CII] emission, going beyond the stellar
continuum as opposed to what is typically observed in the early Universe where
measurements can be affected by the poor sensitivity and faintness of dust
emission in the CGM of high-$z$ galaxies. We attribute the extended [CII] and
dust continuum emission to the presence of galactic outflows. As local dwarf
galaxies are considered analogs of primordial sources, we expect that
comparable feedback processes can be at the origin of the observed [CII] halos
at $z>4$, dominating over other possible formation mechanisms.
