Aims. We present the Lensed Lyman-Alpha MUSE Arcs Sample (LLAMAS) selected from MUSE and HST observations of 17 lensing clusters. The sample consists of 603 continuum-faint (−23 < MUV < −14) lensed Lyman-α emitters (producing 959 images) with secure spectroscopic redshifts between 2.9 and 6.7. Combining the power of cluster magnification with 3D spectroscopic observations, we were able to reveal the resolved morphological properties of 268 Lyman-α emitters.Methods. We used a forward-modeling approach to model both Lyman-α and rest-frame UV continuum emission profiles in the source plane and measure spatial extent, ellipticity, and spatial offsets between UV and Lyman-α emission.Results. We find a significant correlation between UV continuum and Lyman-α spatial extent. Our characterization of the Lyman-α halos indicates that the halo size is linked to the physical properties of the host galaxy (SFR, Lyman-α equivalent width, Lyman-α line FWHM). We find that 48% of Lyman-α halos are best fit by an elliptical emission distribution with a median axis ratio of q = 0.48. We observe that 60% of galaxies detected both in UV and Lyman-α emission show a significant spatial offset (ΔLyα − UV). We measure a median offset of ΔLyα − UV = 0.58 ± 0.14 kpc for the entire sample. By comparing the spatial offset values with the size of the UV component, we show that 40% of the offsets could be due to star-forming sub-structures in the UV component, while the larger offsets (60%) are more likely due to greater-distance processes such as scattering effects inside the circumgalactic medium or emission from faint satellites or merging galaxies. Comparisons with a zoom-in radiative hydrodynamics simulation of a typical Lyman-α emitting galaxy show a very good agreement with LLAMAS galaxies and indicate that bright star-formation clumps and satellite galaxies could produce a similar spatial offset distribution.