  The radio-quiet quasar Q2059-360 at redshift $z=3.08$ is known to be close to
a small Lyman $\alpha$ blob (LAB) and to be absorbed by a proximate damped
Ly$\alpha$ (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$\alpha$ 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$\alpha$ line width. Upper limits are set to the flux of the N V $\lambda
1238-1242$, C IV $\lambda 1548-1551$, He II $\lambda 1640$, and C III] $\lambda
1548-1551$ 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.
