In number theory there are $p$-adic Banach spaces and $p$-adic Banach algebras (e.g., the Tate algebras), and more generally there is the whole subject of $p$-adic functional analysis. Applications of $p$-adic functional analysis go back to the 1960s in work of Dwork (as explained systematically by Serre, go here for an English translation by Jay Pottharst) on the Weil conjectures. For more recent work see here, here, and here. There is a book on this subject by Schneider.
While the Hahn-Banach theorem for real Banach spaces carries over to Banach spaces over $\mathbf Q_p$, examples of duality in the $p$-adic setting look surprising from the viewpoint of classical analysis: while the dual of $c_0(\mathbf R)$ is $\ell^1(\mathbf R)$, the dual of $c_0(\mathbf Q_p)$ is $\ell^\infty(\mathbf Q_p)$. This is due to the different nature of convergence of series in $\mathbf R$ and $\mathbf Q_p$: in $\mathbf R$ the pairing $\langle \{a_n\},\{b_n\}\rangle = \sum_{n \geq 1} a_nb_n$ if $a_n \rightarrow 0$ makes sense if $\sum |b_n| < \infty$ but not generally if $\{b_n\}$ is merely bounded (e.g., $a_n = 1/n$ and $b_n = 1$), while in $\mathbf Q_p$ the same pairing for $a_n \rightarrow 0$ does make sense if $\{b_n\}$ is bounded (for $b_n = 1$ this is the fact that a series of $p$-adic numbers converges if the general term tends to $0$).