We study the projected spatial offset between the ultraviolet continuum and Lya 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 Lya properties of galaxies observed in slit spectroscopy. While we find that ~40% of galaxies in our sample show significant projected spatial offsets, |Delta_Lya-UV_|, we find a relatively modest average (median) projected offset of |Delta_Lya-UV_|=0.61+/-0.08 proper kpc for the entire sample. A small fraction of our sample, ~10%, 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_Lya-UV with redshift. In our sample, UV-bright galaxies (with a median L_UV_/L*_UV_=0.67) showed offsets a factor of three greater than their fainter counterparts (median L_UV_/L*_UV_=0.10), 0.89 +/-0.18 vs. 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 Lya fluorescence and/or backscattering from nearby or outflowing gas. The reduction in the Lya flux due to offsets was quantified. It was found that the differential loss of Lya photons for galaxies with average offsets is not, if corrected for, a limiting factor for all but the narrowest slit widths (<0.4"). 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" were employed, such as those planned for James Webb Space Telescope/NIRSpec observations.