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Cembranos and Freniche proved that for every two infinite compact Hausdorffspaces $X$ and $Y$ the Banach space $C(X\times Y)$ of continuous real-valuedfunctions on $X\times Y$ endowed with the supremum norm contains a complementedcopy of the Banach space $c_{0}$. We extend this theorem to the class of$C_p$-spaces, that is, we prove that for all infinite Tychonoff spaces $X$ and$Y$ the space $C_{p}(X\times Y)$ of continuous functions on $X\times Y$ endowedwith the pointwise topology contains either a complemented copy of$\mathbb{R}^{\omega}$ or a complemented copy of the space$(c_{0})_{p}=\{(x_n)_{n\in\omega}\in \mathbb{R}^\omega\colon x_n\to 0\}$, bothendowed with the product topology. We show that the latter case holds alwayswhen $X\times Y$ is pseudocompact. On the other hand, assuming the ContinuumHypothesis (or even a weaker set-theoretic assumption), we provide an exampleof a pseudocompact space $X$ such that $C_{p}(X\times X)$ does not contain acomplemented copy of $(c_{0})_{p}$.  As a corollary to the first result, we show that for all infinite Tychonoffspaces $X$ and $Y$ the space $C_{p}(X\times Y)$ is linearly homeomorphic to thespace $C_{p}(X\times Y)\times\mathbb{R}$, although, as proved earlier byMarciszewski, there exists an infinite compact space $X$ such that $C_{p}(X)$cannot be mapped onto $C_{p}(X)\times\mathbb{R}$ by a continuous linearsurjection. This provides a positive answer to a problem of Arkhangel'ski forspaces of the form $C_p(X\times Y)$.  Another corollary asserts that for every infinite Tychonoff spaces $X$ and$Y$ the space $C_{k}(X\times Y)$ of continuous functions on $X\times Y$ endowedwith the compact-open topology admits a quotient map onto a space isomorphic toone of the following three spaces: $\mathbb{R}^\omega$, $(c_{0})_{p}$ or$c_{0}$.

On complemented copies of the space $c_0$ in spaces $C_p(X\times Y)$