  We study a one-dimensional two-component atomic Fermi gas with an infinite
intercomponent contact repulsion. It is found that adding an attractive
resonant odd-wave interaction breaking the rotational symmetry one can make the
ground state ferromagnetic. A promising system for the observation of this
itinerant ferromagnetic state is a 1D gas of $^{40}$K atoms, where 3D $s$-wave
and $p$-wave Feshbach resonances are very close to each other and the 1D
confinement significantly reduces the inelastic decay.
