  We report on the dramatic effect of random point defects, produced by proton
irradiation, on the superfluid density $\rho_{s}$ in superconducting
Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ single crystals. The magnitude of the
suppression is inferred from measurements of the temperature-dependent magnetic
penetration depth $\lambda(T)$ using magnetic force microscopy. Our findings
indicate that a radiation dose of 2$\times10^{16}$cm$^{-2}$ produced by 3 MeV
protons results in a reduction of the superconducting critical temperature
$T_{c}$ by approximately 10%. % with no appreciable change in the slope of the
upper critical fields. In contrast, $\rho_{s}(0)$ is suppressed by
approximately 60%. This break-down of the Abrikosov-Gorkov theory may be
explained by the so-called "Swiss cheese model", which accounts for the spatial
suppression of the order parameter near point defects similar to holes in Swiss
cheese. Both the slope of the upper critical field and the penetration depth
$\lambda(T/T_{c})/\lambda(0)$ exhibit similar temperature dependences before
and after irradiation. This may be due to a combination of the highly
disordered nature of Ca$_{0.5}$Na$_{0.5}$Fe$_2$As$_2$ with large intraband and
simultaneous interband scattering as well as the $s^\pm$-wave nature of short
coherence length superconductivity.
