We study the projected spatial offset between the ultraviolet continuum and Ly α emission for 65 lensed and unlensed galaxies in the Epoch of Reionization (5 ≤ z ≤ 7), the first such study at these redshifts, in order to understand the potential for these offsets to confuse estimates of the Ly α properties of galaxies observed in slit spectroscopy. While we find that ∼40 per cent of galaxies in our sample show significant projected spatial offsets ($|\Delta {\rm {Ly}\alpha -\rm {UV}}|$), we find a relatively modest average projected offset of $|\widetilde{\Delta }{\rm {Ly}\alpha -\rm {UV}}|$ = 0.61 ± 0.08 proper kpc for the entire sample. A small fraction of our sample, ∼10 per cent, exhibit offsets in excess of 2 proper kpc, with offsets seen up to ∼4 proper kpc, sizes that are considerably larger than the effective radii of typical galaxies at these redshifts. An internal comparison and a comparison to studies at lower redshift yielded no significant evidence of evolution of $|\Delta {\rm {Ly}\alpha -\rm {UV}}|$ with redshift. In our sample, ultraviolet (UV)-bright galaxies ($\widetilde{L{\mathrm{ UV}}}/L{\ast }{\mathrm{ UV}}=0.67$) showed offsets a factor of three greater than their fainter counterparts ($\widetilde{L{\mathrm{ UV}}}/L{\ast }{\mathrm{ UV}}=0.10$), 0.89 ± 0.18 versus 0.27 ± 0.05 proper kpc, respectively. The presence of companion galaxies and early stage merging activity appeared to be unlikely causes of these offsets. Rather, these offsets appear consistent with a scenario in which internal anisotropic processes resulting from stellar feedback, which is stronger in UV-brighter galaxies, facilitate Ly α fluorescence and/or backscattering from nearby or outflowing gas. The reduction in the Ly α flux due to offsets was quantified. It was found that the differential loss of Ly α photons for galaxies with average offsets is not, if corrected for, a limiting factor for all but the narrowest slit widths (<0.4 arcsec). However, for the largest offsets, if they are mostly perpendicular to the slit major axis, slit losses were found to be extremely severe in cases where slit widths of ≤1 arcsec were employed, such as those planned for James Webb Space Telescope/NIRSpec observations.