  Metal-organic frameworks (MOFs), which are self-assemblies of metal ions and
organic ligands, provide a tunable platform to search a new state of matter. A
two-dimensional (2D) perfect kagome lattice, whose geometrical frustration is a
key to realizing quantum spin liquids, has been formed in the ${\pi}$-${d}$
conjugated 2D MOF [Cu$_{3}$(C$_{6}$S$_{6}$)]$_{n}$ (Cu-BHT). The recent
discovery of its superconductivity with a critical temperature $T_{\rm c}$ of
0.25\,kelvin raises fundamental questions about the nature of electron pairing.
Here, we show that Cu-BHT is a strongly correlated unconventional
superconductor with extremely low superfluid density. A nonexponential
temperature dependence of superfluid density is observed, indicating the
possible presence of superconducting gap nodes. The magnitude of superfluid
density is much smaller than those in conventional superconductors, and follows
the Uemura's relation of strongly correlated superconductors. These results
imply that the unconventional superconductivity in Cu-BHT originates from
electron correlations related to spin fluctuations of kagome lattice.
