Does P = NP imply NP being a strict subset of PSPACE? Does $\textbf{P} = \textbf {NP}$ imply that $\textbf{NP} \subsetneq \textbf{PSPACE}$? Or, for a slightly stronger result, does it imply that $\textbf{NL} = \textbf P$?
No.  It is possible (as far as we know) that $\textbf{P} = \textbf{NP} = \textbf{PSPACE}$.

If $\textbf{P} = \textbf{NP}$, the polynomial hierarchy collapses, i.e., $\textbf{P} = \textbf{PH}$.  See also Can one amplify P=NP beyond P=PH? for an attempt to understand the limits of the implications of $\textbf{P} = \textbf{NP}$, and see Why doesn&#39;t P=NP imply P=AP (i.e. P=PSPACE)? for information about why it seems hard to derive the implication $\textbf{P} = \textbf{PSPACE}$.