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 deep observations (27 hours) of the Hubble Deep Field South from 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* = 8 x 10^9 Msun, star formation rate SFR = 77 Msun/yr, and star formation rate surface brightness 1.6 Msun/kpc^2 within the [OII] half-light radius R_1/2,[OII] = 2.76 +- 0.17 kpc. From a component of the strong resonant MgII and FeII absorptions at -350 km/s, we infer a mass outflow rate that is comparable to the star formation rate. We detect non-resonant FeII* emission, at lambda 2626, 2612, 2396, and 2365, at 1.2-2.4-1.5-2.7 x 10^-18 egs s-1 cm-2 respectively. These flux ratios are consistent with the expectations for optically thick gas. By combining the four non-resonant FeII* emission lines, we spatially map the FeII* emission from an individual galaxy for the first time. The FeII* emission has an elliptical morphology that is roughly aligned with the galaxy minor kinematic axis, and its integrated half-light radius R_1/2,FeII* = 4.1 +- 0.4 kpc is 50% larger than the stellar continuum (R_1/2,* = 2.34 +- 0.17 kpc) or the [OII] nebular line. Moreover, the FeII* emission shows a blue wing extending up to -400 km/s, 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.
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