A space \(X\) is strongly dominated by a space \(Y\) if there exists a continuous surjection \(f:Y \to X\) and a compact cover \(\mathcal F\) of \(X\) such that for any compact set \(E\subset X\), there is a compact set \(K\subset Y\) with \(E\subset f(K)\). Answering a question from [\textit{D. Guerrero Sánchez} and the author, J. Math. Anal. Appl. 454, No. 2, 533--541 (2017; Zbl 1394.54001)], the author proves that if \(X\) is a Lindelöf \(\Sigma\)-space and the function space \(C_p(X,[0,1])\) is strongly dominated by a second countable space, then \(X\) is countable. On the other hand, Martin's Axiom MA implies the existence of a countable space \(Z\) strongly dominating \(C_p(X)\) for an uncountable space \(X\). Also, assuming MA, strong domination by a countable space of \((X \times X) \setminus \Delta_X\) for a compact space \(X\) need not imply metrizability of \(X\). Twelve interesting open problems are posed.