Non-universal families of separable Banach spaces

We prove that if $C$ is a family of separable Banach spaces which is analytic with respect to the Effros-Borel structure and none member of $C$ is isometrically universal for all separable Banach spaces, then there exists a separable Banach space with a monotone Schauder basis which is isometrically universal for $C$ but still not for all separable Banach spaces. We also establish an analogous result for the class of strictly convex spaces.