We present the detection of Ly α, [O iii], and H α emission associated with an extremely strong damped Lyman-α (DLA) system (N(H   i) = 10^22.10 cm^-2) at z = 2.207 towards the quasar SDSS J113520.39 − 001053.56. This is the largest H i column density ever measured along a quasi-stellar object (QSO) line of sight, though typical of those often found in DLAs associated to gamma-ray bursts (GRBs). This absorption system can also be classified as an ultra-strong Mg ii system with \hbox{} Å. The mean metallicity of the gas ( [Zn/H]  =  −1.1) and dust depletion factors ( [Zn/Fe]  = 0.72,  [Zn/Cr]  = 0.49) are consistent with (and only marginally larger than) the mean values found in the general QSO-DLA population. The [O iii]-Hα emitting region has a very small impact parameter with respect to the QSO line of sight, b ≈ 0.1′′ (0.9 kpc proper distance), and is unresolved. From the H α line, we measure a significant star formation rate (SFR) ≈ 25   M_⊙ yr^-1 (uncorrected for dust). The shape of the Ly α line is double-peaked, which is the signature of a resonant scattering of Ly α photons, and the Ly α emission is spatially extended. More strikingly, the blue and red Ly α peaks arise from distinct regions extended over a few kpc on either side of the star-forming region. We propose that this is the consequence of a Ly α transfer in outflowing gas. The presence of starburst-driven outflows is also in agreement with the high SFR together with the small size and low mass of the galaxy (M_vir ~ 10¹⁰   M_⊙). By placing constraints on the stellar UV continuum luminosity of the galaxy, we estimate an age of at most a few 10⁷ yr, again consistent with a recent starburst scenario. We interpret these data as the observation of a young, gas-rich, compact starburst galaxy, from which material is expelled through collimated winds powered by the vigorous star formation activity. We substantiate this picture by modelling the radiative transfer of Ly α photons in the galactic counterpart. Though our model (a spherical galaxy with bipolar outflowing jets) is a simplistic representation of the true gas distribution and velocity field, the agreement between the observed and simulated properties is particularly good (spectral shape and width of the Lyman-α emission, spatial configuration, escape fraction as well as absorption kinematics, H i column density, and dust reddening). Finally, we propose that selecting DLAs with very high H i column densities may be an efficient way of detecting star-forming galaxies at small impact parameters from the background QSO lines of sight.