  The extended Ly$\alpha$ haloes (LAHs) have been found to be prevalent around
high-redshift star-forming galaxies. However, the origin of the LAHs is still a
subject of debate. Spatially resolved analysis of Ly$\alpha$ profiles provides
an important diagnostic. We analyse the average spatial extent and spectral
variation of the circumgalactic LAHs by stacking a sample of 155 Ly$\alpha$
emitters (LAEs) at redshift $3<z<4$ in the MUSE Extremely Deep Field. Our
analysis reveals that, with respect to the Ly$\alpha$ line of the target LAE,
the peak of the Ly$\alpha$ line at large distances becomes increasingly more
blueshifted up to a projected distance of 60 kpc ($\approx 3 \times$ virial
radius), with a velocity offset of $\approx$ 250 km/s. This trend is evident in
both the mean and median stacks, suggesting that it is a general property of
our LAE sample, which typically has a Ly$\alpha$ luminosity $\mathrm{\approx
10^{41.1} erg\,s^{-1}}$. However, due to the absence of systemic redshift data,
it remains unclear whether the Ly$\alpha$ line peak at large projected
distances is less redshifted compared to the inner regions or truly blueshifted
with respect to the systemic velocity. We explore various scenarios to explain
the large-scale kinematics of the Ly$\alpha$ line.
