Fix an integer $n \ge 5$. Let $\mathcal{V}$ be a countable collection of closed subvarieties of $\mathbb{P}^n_{\mathbb{C}}$ of codimension at least $2$. Choose a point $p \in \mathbb{P}^n$. Does there exist a curve $C$ (affine or projective) containing the point $p$ and not intersecting any subvariety $V \in \mathcal{V}$ away from $p$ i.e., for any $V \in \mathcal{V}$, $C \cap V$ is either $p$ or $\emptyset$?

EDIT The underlying field is $\mathbb{C}$.