We examine in quantitative detail the effects of long-range crystalline order on the collective diffusion of an interstitial species that is dissolved in a coherent, binary hard-sphere crystal. A linear constitutive law relating the flux of diffusing interstitials to the chemical driving force for diffusion is derived within the framework of linear-response theory. We use this law to argue that, by contrast with tracer diffusion, collective diffusion in coherent, deformable crystalline solids may be spatially nonlocal (i.e., non-Fickian) in some instances due to effective elastic interactions among the diffusing interstitials. We demonstrate this by performing a molecular-dynamics simulation of interstitial diffusion in a binary hard-sphere crystal. A diffusion response function as a function of orientation in reciprocal space is calculated from the simulation. We find that the limit of the diffusion response function as the wave vector tends to zero remains a function of orientation in reciprocal space, implying that the equation that describes the diffusion of the interstitial species is spatially nonlocal. Our results are consistent with earlier predictions by Cahn concerning the effects of compositional strain on collective diffusion in coherent crystalline solids.