Silverman has discussed the problem of bounding the Mordell–Weil ranks of elliptic curves over towers of function fields (J. Algebraic Geom. 9 (2000), 301–308; J. reine. angew. Math. 577 (2004), 153–169). We first prove generalizations of the theorems of Silverman by a different method, allowing non-abelian Galois groups and removing the dependence on Tate's conjectures. We then prove some theorems about the growth of Mordell–Weil ranks in towers of function fields whose Galois groups are $p$-adic Lie groups; a natural question is whether the Mordell–Weil rank is bounded in such a tower. We give some Galois-theoretic criteria which guarantee that certain curves ${\mathcal{E}}/{\mathbb{Q}}(t)$ have finite Mordell–Weil rank over ${\mathbb{C}}(t^{p^{-\infty}})$, and show that these criteria are met for elliptic K3 surfaces whose associated Galois representations have sufficiently large image.