  Among the iron-based pnictide superconductors the material KFe$_2$As$_2$ is
unusual in that its Fermi surface does not consist of quasi-nested electron and
hole pockets. Here we report measurements of the temperature dependent London
penetration depth of very clean crystals of this compound with residual
resistivity ratio $>1200$. We show that the superfluid density at low
temperatures exhibits a strong linear-in-temperature dependence which implies
that there are line nodes in the energy gap on the large zone-centered hole
sheets. The results indicate that KFe$_2$As$_2$ is an unconventional
superconductor with strong electron correlations.
