The in-plane resistivity $rho$ and thermal conductivity $kappa$ of extremely overdoped KFe$_2$As$_2$ ($T_c$ = 3.0 K) single crystal were studied. It is found that $rho sim T^{1.5}$ at low temperature, a typical non-Fermi liquid behavior of electrons scattered by antiferromagnetic spin fluctuations. In zero field, we observed a large residual linear term $kappa_0/T$, about one third of the normal-state value. In low magnetic fields, $kappa_0/T(H)$ increases very fast. Such a behavior of $kappa_0/T$ mimics the d-wave cuprate superconductors, therefore provides clear evidence for nodes in the superconducting gap of KFe$_2$As$_2$. Based on the Fermi surface topology of KFe$_2$As$_2$, it is believed that the dominant intraband pairing via antiferromagnetic spin fluctuations results in the unconventional superconducting gap with nodes.
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