Enriched purity and presentability in Banach spaces

The category $Ban$ of Banach spaces and linear maps of norm $\le 1$ is locally ${\mathrm{\aleph }}_1$-presentable but not locally finitely presentable. We prove, however, that $Ban$ is locally finitely presentable in the enriched sense over complete metric spaces. Moreover, in this sense, pure morphisms are just ideals of Banach spaces. We characterize classes of Banach spaces approximately injective to sets of morphisms having finite-dimensional domains and separable codomains.