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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 work of Dwork (as explained systematically by Serre) in the 1960s on the Weil conjectures. For more recent work see here and here, and here. There is a book on this subject by Schneider.