  We report the discovery of diffuse extended Ly-alpha emission from redshift
3.1 to 4.5, tracing cosmic web filaments on scales of 2.5-4 comoving Mpc. These
structures have been observed in overdensities of Ly-alpha emitters in the MUSE
Extremely Deep Field, a 140 hour deep MUSE observation located in the Hubble
Ultra Deep Field. Among the 22 overdense regions identified, 5 are likely to
harbor very extended Ly-alpha emission at high significance with an average
surface brightness of $\mathrm{5 \times 10^{-20} erg s^{-1} cm^{-2}
arcsec^{-2}}$. Remarkably, 70% of the total Ly-alpha luminosity from these
filaments comes from beyond the circumgalactic medium of any identified
Ly-alpha emitters. Fluorescent Ly-alpha emission powered by the cosmic UV
background can only account for less than 34% of this emission at z$\approx$3
and for not more than 10% at higher redshift. We find that the bulk of this
diffuse emission can be reproduced by the unresolved Ly-alpha emission of a
large population of ultra low luminosity Ly-alpha emitters ($\mathrm{<10^{40}
erg s^{-1}}$), provided that the faint end of the Ly-alpha luminosity function
is steep ($\alpha \lessapprox -1.8$), it extends down to luminosities lower
than $\mathrm{10^{38} - 10^{37} erg s^{-1}}$ and the clustering of these
Ly-alpha emitters is significant (filling factor $< 1/6$). If these Ly-alpha
emitters are powered by star formation, then this implies their luminosity
function needs to extend down to star formation rates $\mathrm{< 10^{-4}
M_\odot yr^{-1}}$. These observations provide the first detection of the cosmic
web in Ly-alpha emission in typical filamentary environments and the first
observational clue for the existence of a large population of ultra low
luminosity Ly-alpha emitters at high redshift.
