  We study the fraction of the intra-cluster light (ICL) formed in-situ in the
three most massive clusters of the TNG50 simulation, with virial masses $\sim
10^{14}$ M$_{\odot}$. We find that a significant fraction of ICL stars
($8\%$-$28\%$) are born in-situ. This amounts to a total stellar mass
comparable to the central galaxy itself. Contrary to simple expectations, only
a sub-dominant fraction of these in-situ ICL stars are born in the central
regions and later re-distributed to more energetic orbits during mergers.
Instead, many in-situ ICL stars form directly hundreds of kiloparsecs away from
the central galaxy, in clouds condensing out of the circum-cluster medium. The
simulations predict a present-date diffuse star formation rate of $\sim$1
$\mathrm{M}_{\odot}$/yr, with higher rates at higher redshifts. The diffuse
star forming component of the ICL is filamentary in nature, extends for
hundreds of kiloparsecs and traces the distribution of neutral gas in the
cluster host halo. We discuss briefly how numerical details of the baryonic
treatment in the simulation may play a role in this result and conclude that a
sensitivity of $1.6 \times 10^{-19} - 2.6 \times 10^{-18}$ erg s$^{-1}$
cm$^{-2}$ arcsec$^{-2}$ in H$_\alpha$ flux (beyond current observational
capabilities) would be necessary to detect this diffuse star-forming component
in galaxy clusters.
