  Let $E$ be an elliptic curve defined over a number field $F$ with good
ordinary reduction at all primes above $p$, and let $F_\infty$ be a finitely
ramified uniform pro-$p$ extension of $F$ containing the cyclotomic
$\mathbb{Z}_p$-extension $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of the
tower, and $F^{(n)}_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of
$F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the
growth of the $\lambda$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as
$n\rightarrow \infty$. This method has its origins in work of A.Cuoco, who
studied $\mathbb{Z}_p^2$-extensions. Refined estimates for growth are proved
that are close to conjectured estimates. The results are illustrated in special
cases.
