  We develop the machinery of indiscernible subspaces in continuous theories of
expansions of Banach spaces, showing that any such theory has an indiscernible
subspace and therefore an indiscernible set. We extend a result of Shelah and
Usvyatsov by showing that a sequence of realizations of a (possibly unstable)
minimal wide type $p$ is a Morley sequence in $p$ if and only if it is the
orthonormal basis of an indiscernible subspace in $p$. We also give an example
showing that minimal wide types do not generally have type-definable
indiscernible subspaces (answering a question of Shelah and Usvyatsov), as well
as an example showing that our result fails for non-minimal wide types, even in
$\omega$-stable theories.
