How do I prove that SPACE($n^{555}$) $\neq$ NP?
According to the space hierarchy theorem, there is a language in $\mathrm{SPACE}(n^{556})$ which is not in $\mathsf{SPACE}(n^{555})$. A padded version of this language will be in $\mathsf{SPACE}(n^{555})$. If the latter is equal to $\mathsf{NP}$, then the original language would also be in $\mathsf{NP} = \mathsf{SPACE}(n^{555})$, since we can implement padding while remaining inside $\mathsf{NP}$.

I'll let you flesh out this argument.