  In this article we describe a multi-layered honeycomb lattice model of
interacting fermions which supports a new kind of parity-preserving skyrmion
superfluidity. We derive the low-energy field theory describing a non-BCS
fermionic superfluid phase by means of functional fermionization. Such
effective theory is a new kind of non-linear sigma model, which we call double
skyrmion model. In the bi-layer case, the quasiparticles of the system
(skyrmions) have bosonic statistics and replace the Cooper-pairs role.
Moreover, we show that the model is also equivalent to a Maxwell-BF theory,
which naturally establishes an effective Meissner effect without requiring a
breaking of the gauge symmetry. Finally, we map effective superfluidity effects
to identities among fermionic observables for the lattice model. This provides
a signature of our theoretical skyrmion superfluidy that can be detected in a
possible implementation of the lattice model in a real quantum system.
