Sure, but of course the Newtonian limit (for a mechanical situation) can be valid only in an inertial frame, where the bodies move with velocities much smaller than the speed of light, i.e., you'll get a good approximation to the relativistic dynamics only in such reference frames.
How do you come to that conclusion? There are both types of currents in nature: A "convection current", i.e., the current due to a single moving charge is of course timelike. In continuum-mechanical notation it's given by $$j^{\mu}=q n c u^{\mu},$$
where ##q## is the charge of the particles making up the fluid, ##n## the particle density as measured in the rest frame of the fluid cell (a scalar), and ##u^{\mu}## the normalized four-velocity (with ##u_{\mu} u^{\mu}=1##, using the (1,-1,-1,-1) signature).
Then there are conduction-current densities in wires, which are space-like. The charge density is close to 0 since there is the positive ion lattice in addition to the negative conduction electrons making up the current.
It's Eq. (2.23) on p. 224, and in my edition of Landau-Lifshitz's vol. 2 it's in Paragraph 24. That's indeed an approximation of the transformation law derived as an expansion in powers of ##1/c##, but indeed this doesn't lead to a transformation group and in this sense is not a consistent Galilean theory. Of course the paper also demonstrates that there is indeed no Galilean electrodynamics which is consistent with the phenomenology anyway.
Well, the question, whether there is a consistent Galilean electrodynamics is of some academic interest, but as the paper shows, it fails to describe the electromagnetic phenomenology right although of course there are good approximations to certain "non-relativistic" situations, e.g., the quasistationary approximations used to derive AC circuit theory.