A $\sim$15 kpc outflow cone piercing through the halo of the blue compact metal-poor galaxy SBS0335-052

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.