The radio-quiet quasar Q2059-360 at redshift z = 3 : 08 is known to be close to a small Lyman ff blob (LAB) and to be absorbed by a proximate damped Ly ff (PDLA) system. Here, we present the Multi Unit Spectroscopic Explorer (MUSE) integral field spectroscopy follow-up of this quasi-stellar object (QSO). Our primary goal is to characterize this LAB in detail by mapping it both spatially and spectrally using the Ly ff line, and by looking for high-ionization lines to constrain the emission mechanism. Combining the high sensitivity of the MUSE integral field spectrograph mounted on the Yepun telescope at ESO-VLT with the natural coronagraph provided by the PDLA, we map the LAB down to the QSO position, after robust subtraction of QSO light in the spectral domain. In addition to confirming earlier results for the small bright component of the LAB, we unveil a faint filamentary emission protruding to the south over about 80 pkpc (physical kpc); this results in a total size of about 120 pkpc. We derive the velocity field of the LAB (assuming no transfer effects) and map the Ly ff line width. Upper limits are set to the flux of the N v lambda 1238 1242, C iv lambda 1548 1551, Heii lambda 1640, and C i i i] lambda 1548 1551 lines. We have discovered two probable Ly alpha emitters at the same redshift as the LAB and at projected distances of 265 kpc and 207 kpc from the QSO; their Ly alpha luminosities might well be enhanced by the QSO radiation. We also find an emission line galaxy at z = 0 : 33 near the line of sight to the QSO. This LAB shares the same general characteristics as the 17 others surrounding radio-quiet QSOs presented previously. However, there are indications that it may be centered on the PDLA galaxy rather than on the QSO.