Superconducting critical field far above the Pauli limit in one-dimensional Li$_{0.9}$Mo$_6$O$_{17}$

The upper critical field $H_{c2}$ of purple bronze Li$_{0.9}$Mo$_6$O$_{17}$ is found to exhibit a large anisotropy, in quantitative agreement with that expected from the observed electrical resistivity anisotropy. With the field aligned along the most conducting axis, $H_{c2}$ increases monotonically with decreasing temperature to a value five times larger than the estimated paramagnetic pair-breaking field. Theories for the enhancement of $H_{c2}$ invoking spin-orbit scattering or strong-coupling superconductivity are shown to be inadequate in explaining the observed behavior, suggesting that the pairing state in Li$_{0.9}$Mo$_6$O$_{17}$ is unconventional and possibly spin-triplet.