  The recent discovery of superconductivity in NaSn$_2$As$_2$ with a van der
Waals layered structure raises immediate questions on its pairing mechanism and
underlying electronic structure. Here, we present measurements of the
temperature-dependent magnetic penetration depth $\lambda(T)$ in single
crystals of NaSn$_2$As$_2$ down to $\sim40$ mK. We find a very long penetration
depth $\lambda (0) = 960$ nm, which is strongly enhanced from the estimate of
first-principles calculations. This enhancement comes from a short mean free
path $\ell \approx 1.7$ nm, indicating atomic scale disorder possibly
associated with the valence-skipping states of Sn. The temperature dependence
of superfluid density is fully consistent with the conventional fully gapped
s-wave state in the dirty limit. These results suggest that NaSn$_2$As$_2$ is
an ideal material to study quantum phase fluctuations in strongly disordered
superconductors with its controllable dimensionality.
