There is a concept of the free Banach space over a metric space.  This a canonical way to embed a metric space into a Banach space and was introduced by Arens and Eells in a paper in the Pacific Journal of Mathematics ("On embedding uniform and topological spaces", vol. 6 (1956), 397-403).  One way to construct it is as a predual of a suitable space of Lipschitz functions on the metric space.