In this paper, we investigate the base-$p$ expansions of putative counterexamples to the $p$-adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base-$p$ expansion is uniformly recurrent. Furthermore, we show that if the base-$p$ expansion of $x$ is a morphic word $τ(φ^ω(a))$ where $φ^ω(a)$ contains a subword of the form $uXuXu$ with $\lim_{n\to\infty}|φ^n(u)|=\infty$, then $x$ satisfies the $p$-adic Littlewood conjecture. In the special case when $p=2$, we show that the conjecture holds for all pure morphic words.