We will present some results on existence of dense functionally countable subspaces in spaces $C_p(X)$. It will be shown, among other things, that there is a consistent example of a scattered Lindel"of $P$-space $X$ for which $C_p(X)$ has no dense functionally countable subspace and that $\mathbb R^{\omega_1}$ has a dense functionally countable subspace of cardinality $\omega_2$ if and only if the Kurepa Hypothesis holds.