We report the discovery of bulk superconductivity (SC) at 6.1 K in a quasi-one-dimensional (Q1D) chromium pnictide K$_2$Cr$_3$As$_3$ which contains [(Cr$_3$As$_3$)$^{2-}$]$_{infty}$ double-walled subnano-tubes with face-sharing Cr$_{6/2}$ (As$_{6/2}$) octahedron linear chains in the inner (outer) wall. The material has a large electronic specific-heat coefficient of 70$sim$75 mJ K$^{-2}$ mol$^{-1}$, indicating significantly strong electron correlations. Signature of non-Fermi liquid behavior is shown by the linear temperature dependence of resistivity in a broad temperature range from 7 to 300 K. Unconventional SC is preliminarily manifested by the estimated upper critical field exceeding the Pauli limit by a factor of three to four. The title compound represents a rare example that possibly unconventional SC emerges in a Q1D system with strong electron correlations.
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