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.
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