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 $q$).