A locally convex space (lcs) \(E\) is said to have an \(\omega^{\omega}\)-base if \(E\) has a neighborhood base \(\{U_{\alpha}:\alpha\in\omega^\omega\}\) at zero such that \(U_{\beta}\subseteq U_{\alpha}\) for all \(\alpha\leq\beta\). The class of lcs with an \(\omega^{\omega}\)-base is large, among others contains all \((LM)\)-spaces (hence \((LF)\)-spaces), strong duals of distinguished Fr\'echet lcs (hence spaces of distributions \(D'(\Omega)\)). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an \(\omega^{\omega}\)-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an \(\omega^{\omega}\)-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space \(\varphi\) endowed with the finest locally convex topology has an \(\omega^\omega\)-base but contains no infinite-dimensional compact subsets. It turns out that \(\varphi\) is a unique infinite-dimensional locally convex space which is a \(k_{\mathbb{R}}\)-space containing no infinite-dimensional compact subsets. Applications to spaces \(C_{p}(X)\) are provided.