  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)
