  We consider the non-equilibrium orbital dynamics of spin-polarized ultracold
fermions in the first excited band of an optical lattice. A specific lattice
depth and filling configuration is designed to allow the $p_x$ and $p_y$
excited orbital degrees of freedom to act as a pseudo-spin. Starting from the
full Hamiltonian for p-wave interactions in a periodic potential, we derive an
extended Hubbard-type model that describes the anisotropic lattice dynamics of
the excited orbitals at low energy. We then show how dispersion engineering can
provide a viable route to realizing collective behavior driven by p-wave
interactions. In particular, Bragg dressing and lattice depth can reduce
single-particle dispersion rates, such that a collective many-body gap is
opened with only moderate Feshbach enhancement of p-wave interactions. Physical
insight into the emergent gap-protected collective dynamics is gained by
projecting the Hamiltonian into the Dicke manifold, yielding a one-axis
twisting model for the orbital pseudo-spin that can be probed using
conventional Ramsey-style interferometry. Experimentally realistic protocols to
prepare and measure the many-body dynamics are discussed, including the effects
of band relaxation, particle loss, spin-orbit coupling, and doping.
