  Chiral $p$-wave superfluids are fascinating topological quantum states of
matter that have been found in the liquid $^3$He-A phase and arguably in the
electronic Sr$_2$RuO$_4$ superconductor. They are shown fundamentally related
to the fractional $5/2$ quantum Hall state which supports fractional exotic
excitations. A common understanding is that such states require spin-triplet
pairing of fermions due to $p$-wave interaction. Here we report by controlled
theoretical approximation that a center-of-mass Wannier $p$-wave chiral
superfluid state can arise from spin-singlet pairing for an $s$-wave
interacting atomic Fermi gas in an optical lattice. Despite a conceptually
different origin, it shows topological properties similar to the conventional
chiral $p$-wave state. These include a non-zero Chern number and the appearance
of chiral fermionic zero modes bounded to domain walls. Several signature
quantities are calculated for the cold atom experimental condition.
