Chiral p-wave superfluids are fascinating topological quantum states of matter that have been found in the liquid He-3-A phase and arguably in the electronic Sr2RuO4 superconductor. They are fundamentally related to the fractional 5/2 quantum Hall state, which supports fractional exotic excitations. Past studies show that they require spin-triplet pairing of fermions by p-wave interaction. Here we report that a p-wave chiral superfluid state can arise from spin-singlet pairing for an s-wave interacting atomic Fermi gas in an optical lattice. This p-wave state is conceptually distinct from all previous conventional p-wave states as it is for the centre-of-mass motion, instead of the relative motion. It leads to spontaneous generation of angular momentum, finite Chern numbers and topologically protected chiral fermionic zero modes bounded to domain walls, all occuring at a higher critical temperature in relative scales. Signature quantities are predicted for the cold atom experimental condition.