Let $\beta_{\bf2} S$ be a compact, totally-disconnected space containing a dense, discrete subspace $S$ such that any function $f:S\to\bf2$ extends to a continuous map $\hat f:\beta_{\bf2} S\to\bf2$, where $\bf2$ is $\{0,1\}$ with the discrete topology. Let $P$ be a compact, totally-disconnected space.
Does any continuous function $g:S\times P\to\bf2$ extend to a continuous function $\hat g:(\beta_{\bf2} S)\times P\to\bf2$?
No, your $\beta_2S$ is $\beta S$, the Čech-Stone compactification of $S$. As an example let $S=\omega$ and $P=\omega+1$ (with order topology, so it is just the converging sequence). Let $g$ be the characteristic function of the diagonal $\Delta=\{\langle n,n\rangle:n\in\omega\}$. It has no continuous extension to $\beta S\times P$ because the closure of $\Delta$ and the closure of the top line $\omega\times\{\omega\}$ intersect (the intersection is $(\beta\omega\setminus\omega)\times\{\omega\}$); at the points in the intersection the extension should assume both values $0$ and $1$.
Addendum (2024-10-06): In Stone-Čech compactifications of products Glicksberg characterized when $\beta X\times\beta Y=\beta(X\times Y)$ (also for arbitrary products): in the non-trivial case, where $X$ and $Y$ are infinite, the product $X\times Y$ should be pseudocompact.
