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.
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