Problems in $\textbf{DSPACE}(\log^2 n)$
There are about $2^{\log^2n}$ possible configurations, and $2^{\log^2n} = n^{\log n}$, which is not polynomial (it's "quasipolynomial"). So, no, you can't claim that $\mathrm{DSPACE}[\log^2 n]\subseteq \mathrm{P}$.