In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Super-properties are also characterized--more commonly?--via finite representability).

What is the name for a property of a Banach space preserved under ultraproducts.

For example, a Banach space $B$ is super-reflexive iff it is reflexive and all its ultrapowers are reflexive. However, super-reflexitivity is not preserved under ultraproducts. Indeed, the spaces $L^p$ for $1<p<\infty$ are super-reflexive, but if $\mathcal{U}$ is a non-principle ultrafilter on $\mathbb{N}$, then the ultraproduct $(\prod_{n=2}^\infty L^n)/\mathcal{U}$ is not reflexive.

Every super-reflexive space is isomorphic (in the Banach space sense) to a $q$-uniformly convex space for $2\leq q <\infty$. Such a space is said to have martingale cotype $q$. The property of having martingale cotype $q$ is preserved under ultraproducts (for a fixed $q$).