The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Thomassen (2018) and Hartke, Jahanbekam and Thomas (2016) proved that $χ(G^2) \leq 7$ if $G$ is a subcubic planar graph. A natural question is whether $χ_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph. Cranston and Kim (2008) showed that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 7. We prove that $χ_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6.