Almost Free Non-Archimedean Banach Spaces and Relation to Large Cardinals

Let $k$ be a complete valuation field. We formulate a free Banach $k$-vector space as a Banach $k$-vector space with an orthonormal Schauder basis, and an almost free Banach $k$-vector space as a non-Archimedean analogue of an almost free Abelian group. As non-Archimedean analogues of the classical facts that an almost free Abelian group is free under the assumption of the ${\mathrm{\aleph }}_1$-strong compactness or the weak compactness of the cardinality, we show that an almost free Banach $k$-vector space is free under similar assumptions.