  We report the finding of a novel pairing state in a newly discovered
superconductor Na$_2$Cr$_3$As$_3$. This material has a noncentrosymmetric
quasi-one-dimensional crystal structure and is superconducting at
$T_{\mathrm{C}}\sim$ 8.0 K. We find that the magnetic penetration depth data
suggests the presence of a nodal line $p_z$-wave pairing state with zero
magnetic moment using transverse-field muon-spin rotation (TF-$\mu$SR)
measurements. The nodal gap observed in Na$_2$Cr$_3$As$_3$ compound is
consistent with that observed in isostructural (K,Cs)$_2$Cr$_3$As$_3$ compounds
using TF-$\mu$SR measurements. The observed pairing state is consistent with a
three-band model spin-fluctuation calculation, which reveals the $S_z=0$
spin-triplet pairing state with the $\sin k_z$ pairing symmetry. The
long-sought search for chiral superconductivity with topological applications
could be aided by such a novel triplet $S_z=0$ $p$-wave pairing state.
