  We consider the possibility that current hints of spatial variations of the
fine structure constant at high redshift, based on VLT/UVES and Keck/HIRES
observations, could be caused by a biased domain wall network described by a
scalar field non-minimally coupled to the electromagnetic field. We show that
in order to be responsible for the reported spatial variations of the fine
structure constant, the fractional contribution of the domain wall network to
the energy density of the Universe should be tightly constrained within the
range $10^{-10} < \Omega_{w0} < 10^{-5}$. We also show that the domain wall
dynamics should be essentially frictionless, so that its characteristic scale
is in the order of the Hubble radius at the present time.
