We study an ultracold gas of fermionic potassium near a p-wave Feshbach resonance in an optical lattice that confines the atoms into one-dimensional (1D) tubes. Using spin-flip spectroscopy, we measure spatial correlations in the gas as a function of interaction strength and investigate the interplay of band excitations, anisotropy, and dipole-dipole interactions. We compare to quasi-1D theory predictions as well as Bethe Ansatz and high temperature calculations for the 1D p-wave contacts.