Over archimedian fields, examples of maps $ f \colon X \to Y $ of Banach spaces with non-closed image are well-known, e.g. the inclusion $ \ell^1 \hookrightarrow \ell^2 $ is such an example (which can be generalized in several directions).
I am looking for an analogous example over a non-archimedian complete field like $ \mathbb{Q}_p $. The vector space norms on $ X $ and $ Y $ should be non-archimedian aswell. Also pointings to the literature are appreciated.
Let $k$ be a non-archimedian field and $r>0$ a positive real number. Define $k\{r^{-1}T\}$ to be the space of all sequences $(x_i\in k,i\in\mathbb{N})$ such that $r^i\cdot \lvert x_i\rvert\to 0$ as $i\to\infty$. This space becomes a Banach space with the Gauß norm $\lVert(x_i,i\in\mathbb{N})\rVert=\max_{i\in\mathbb{N}}r^i\cdot\lvert x_i\rvert$. If $r\le s$ then the inclusion $k\{s^{-1}T\}\to k\{r^{-1}T\}$ is a contractive map of Banach spaces with dense image which won't be surjective in general.
Note that $ k\{r^{-1}\}$ with its Gauß norm is in fact a Banach algebra with the Cauchy product. In non-archimedian analytic geometry it plays the role of the algebra of analytic functions on the closed unit disk of radius $r$. The above map can be understood as the restriction of functions on the closed $s$-disk to the closed $r$-disk.