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Emission signatures from galactic winds provide an opportunity to directly map the outflowing gas, but this is traditionally challenging because of the low surface brightness. Using very deep observations (27 h) of the Hubble Deep Field South with the Multi Unit Spectroscopic Explorer (MUSE) instrument, we identify signatures of an outflow in both emission and absorption from a spatially resolved galaxy at z = 1.29 with a stellar mass M-star = 8 x 10(9) M-circle dot, star formation rate SFR = 77(-25)(+40) M-circle dot yr(-1), and star formation rate surface brightness Sigma(SFR) = 1.6 M-circle dot kpc(-2) within the [OII] lambda lambda 3727, 3729 half-light radius R-1/2, ([OII]) = 2.76 +/- 0.17 kpc. From a component of the strong resonant Mg II and Fe II absorptions at -350 km s(-1), we infer a mass outflow rate that is comparable to the star formation rate. We detect non-resonant Fe II* emission, at lambda 2365, lambda 2396, lambda 2612, and lambda 2626, at 1.2-2.4-1.5-2.7 x 10-(18) erg s(-1) cm(-2) respectively. The flux ratios are consistent with the expectations for optically thick gas. By combining the four non-resonant Fe II* emission lines, we spatially map the Fe II* emission from an individual galaxy for the first time. The Fe II* emission has an elliptical morphology that is roughly aligned with the galaxy minor kinematic axis, and its integrated half-light radius, R-1/2, (Fe II*) = 4.1 +/- 0.4 kpc, is 70% larger than the stellar continuum (R-1/2,(star) similar or equal to 2.34 +/- 0.17) or the [O II] nebular line. Moreover, the Fe II* emission shows a blue wing extending up to -400 km s(-1), which is more pronounced along the galaxy minor kinematic axis and reveals a C-shaped pattern in a p - v diagram along that axis. These features are consistent with a bi-conical outflow.