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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<M_UV<-14) lensed Lyman-alpha emitters (producing 959 images) with spectroscopic redshifts between 2.9 and 6.7. Combining the power of cluster magnification with 3D spectroscopic observations, we are able to reveal the resolved morphological properties of 268 Lyman-alpha emitters. We use a forward modelling approach to model both Lyman-alpha and rest-frame UV continuum emission profiles in the source plane and measure spatial extent, ellipticity and spatial offsets between UV and Lyman-alpha emission. We find a significant correlation between UV continuum and Lyman-alpha spatial extent. Our characterization of the Lyman-alpha haloes indicates that the halo size is linked to the physical properties of the host galaxy (SFR, Lyman-alpha EW and Lyman-alpha line FWHM). We find that 48% of Lyman-alpha haloes are best-fitted 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-alpha emission show a significant spatial offset (Delta). We measure a median offset of Delta= 0.58 \pm 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 are more likely due to larger 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-alpha emitting galaxy show a good agreement with LLAMAS galaxies and indicate that bright star-formation clumps and satellite galaxies could produce a similar spatial offsets distribution. (abridged)