  We propose an ultracold-atom setting where a fermionic superfluidity with
attractive s-wave interaction is uploaded in a non-Hermitian Lieb optical
lattice. The existence of a real-energy flat band solution is revealed. We show
that the interplay between the skin effect and flat-band localization leads to
exotic localization properties. We develop a multiband mean-field description
of this system and use both order parameters and superfluid weight to describe
the phase transition. A relation between the superfluid weight and
non-Hermitian quantum metric of the quantum states manifold is built. We find
non-monotone criticality depending on the non-Hermiticity, and the skin effect
prominently enhances the phase coherence of the pairing field, suggesting
ubiquitous critical behavior of the non-Hermitian fermionic superfluidity.
