  We study a two-component mixture of fermionic dipoles in two dimensions at
zero temperature, interacting via a purely repulsive $1/r^3$ potential. This
model can be realized with ultracold atoms or molecules, when their dipole
moments are aligned in the confinement direction orthogonal to the plane. We
characterize the unpolarized mixture by means of the Diffusion Monte Carlo
technique. Computing the equation of state, we identify the regime of validity
for a mean-field theory based on a low-density expansion and compare our
results with the hard-disk model of repulsive fermions. At high density, we
address the possibility of itinerant ferromagnetism, namely whether the ground
state can be fully polarized in the fluid phase. Within the fixed-node
approximation, we show that the accuracy of Jastrow-Slater trial wave
functions, even with the typical two-body backflow correction, is not
sufficient to resolve the relevant energy differences. By making use of the
iterative-backflow improved trial wave functions, we observe no signature of a
fully-polarized ground state up to the freezing density.
