In this work we report deep MUSE observations of a broad absorption line (BAL) quasar at z ∼ 5, revealing a Ly α nebula with a maximum projected linear size of ∼60 kpc around the quasar (down to our 2σ SB limit per layer of ∼9×10−19ergs−1cm−2arcsec−2 for a 1 arcsec2 aperture). After correcting for the cosmological surface brightness dimming, we find that our nebula, at z ∼ 5, has an intrinsically less extended Ly α emission than nebulae at lower redshift. However, such a discrepancy is greatly reduced when referring to comoving distances, which take into account the cosmological growth of dark matter (DM) haloes, suggesting a positive correlation between the size of Ly α nebulae and the sizes of DM haloes/structures around quasars. Differently from the typical nebulae around radio-quiet non-BAL quasars, in the inner regions (∼10 kpc) of the circumgalactic medium of our source, the velocity dispersion of the Ly α emission is very high (FWHM > 1000 km s−1), suggesting that in our case we may be probing outflowing material associated with the quasar.