For $x \in \mathbb{Q}_p^n$, $\|x\|_p = \max_{1 \leq i \leq n} |x_i|_p$ where $| \cdot |_p$ is the $p$-adic absolute value.
Let $a \in \mathbb{Z}^n$, $a \neq 0$. How can we find $a' \in \mathbb{Q}_p^n$ such that $$ a^T a' = \sum_{i} a_i a_i' = 1 \quad \text{ and } \quad \|a\|_p \|a'\|_p \leq 1? $$
If $\mathbb{Q}_p$ is replaced by $\mathbb{R}$ and $\| \cdot \|_p$ is replaced by the Euclidean norm, we could take $a' = \dfrac{1}{a^T a} a$.
This question came up in thinking about this question: In a $p$-adic vector space, closest point on a plane to a given point?