 Solution

Add constants $c_1,\dots ,c_n$ and form $T\cup \{\phi (c):\phi \in p\}$. Every finite set has a model by consistency of the type. The compactness theorem gives a model $\mathcal{M}$ realizing the whole expanded theory, so $c^{\mathcal{M}}$ realizes $p$.

Apply the downward Löwenheim-Skolem theorem to the countable expanded language, retaining these finitely many named elements. The resulting countable elementary substructure still satisfies every formula of $p$ at the same tuple. Its reduct to the original language is a countable model of $T$ realizing $p$. Since $T$ has no finite models, this model is countably infinite.