  We study the Singlet-Doublet dark matter model in the regime of feeble
couplings, where the dark matter abundance is obtained via the freeze-in
mechanism. As a consequence of the small couplings, the heavier particles in
the model are long-lived with decay length at typical scales of collider
experiments. We analyse the collider signatures of the model, characterised by
displaced $h$ and $Z$ bosons plus missing momentum, employing current LHC
searches for displaced vertices and missing energy to significantly constrain
the parameter space of the model. We also take into account the cosmological
bounds relevant for our light dark matter candidate arising from Lyman-$\alpha$
forest constraints. Our analysis emphasises the interplay between displaced
signatures at the LHC and cosmology for dark matter candidates whose relic
abundance is obtained through the freeze-in mechanism.
