  Context: Outflows from low-mass star-forming galaxies are a fundamental
ingredient for models of galaxy evolution and cosmology.
  Aims: The onset of kpc-scale ionised filaments in the halo of the metal-poor
compact dwarf SBS 0335-052E was previously not linked to an outflow. We here we
investigate whether these filaments provide evidence for an outflow.
  Methods: We obtained new VLT/MUSE WFM and deep NRAO/VLA B-configuration 21cm
data of the galaxy. The MUSE data provide morphology, kinematics, and emission
line ratios H$\beta$/H$\alpha$ and [\ion{O}{iii}]$\lambda5007$/H$\alpha$ of the
low surface-brightness filaments, while the VLA data deliver morphology and
kinematics of the neutral gas in and around the system. Both datasets are used
in concert for comparisons between the ionised and the neutral phase.
  Results: We report the prolongation of a lacy filamentary ionised structure
up to a projected distance of 16 kpc at $\mathrm{SB}_\mathrm{H\alpha} =
1.5\times10^{-18}$erg s$^{-1}$ cm$^{-2}$arcsec$^{-2}$. The filaments exhibit
unusual low H$\alpha$/H$\beta \approx 2.4$ and low [\ion{O}{iii}]/H$\alpha \sim
0.4 - 0.6$ typical of diffuse ionised gas. They are spectrally narrow ($\sim
20$ km s$^{-1}$) and exhibit no velocity sub-structure. The filaments extend
outwards of the elongated \ion{H}{I} halo. On small scales the $N_\mathrm{HI}$
peak is offset from the main star-forming sites. Morphology and kinematics of
\ion{H}{I} and \ion{H}{II} reveal how star-formation driven feedback interacts
differently with the ionised and the neutral phase.
  Conclusions: We reason that the filaments are a large scale manifestation of
star-formation driven feedback, namely limb-brightened edges of a giant outflow
cone that protrudes through the halo of this gas-rich system. A simple toy
model of such a conical-structure is found to be commensurable with the
observations.
