I am trying to solve the following problem and I got stuck.
Let $k$ be a locally compact complete non-archimedean field and let $X$ be a $k$-analytic submanifold (everywhere of dimension $d$) of the unit ball $B\subset k^n$ (in other words $B$ is just $\mathcal O_k^n$). For each $n$ there is a reduction morphism $r_n:B \to (\mathcal O_k/\mathfrak m_k^n)^N$, denote by $X_n$ the image $r(X)$ and denote the cardinality #$X_n$ by $c_n$. Then there are constants $m, A$ such that $c_n=Ap^{dn}$ for each $n>m$.
I think that it should boil down to some sort of Hensel's Lemma, but I cannot rigorously solve this problem. Any hints?