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 $q$**.  <strike>The property of having martingale cotype $q$ is preserved under ultraproducts (for a fixed $q$).</strike>

**Update 1 (2015-09-27)**: As Bill Johnson pointed out, having $q$-martingale cotype is a bad example, since it is not actually preserved under ultraproducts.

For a better example, being a uniformly convex Banach space with modulus $\delta(\varepsilon)$ is preserved under ultraproducts.  [Proof: Let $\{B_i\}_{i\in I}$ be uniformly convex spaces with common modulus $\delta(\varepsilon)$.  Let $x=(x_i)_\mathcal{U}$ and $y=(y_i)_\mathcal{U}$ be elements in the interior of the unit ball of the ultraproduct $B = \prod_{i\in I} B_i/\mathcal{U}$ with $\|x-y\|_B>\varepsilon$.  Then $\mathcal{U}$-a.s. $\|x_i\|_{B_i}, \|y_i\|_{B_i} \leq 1$ and $\|x_i - y_i\|_{B_i} \geq \varepsilon$.  By uniform convexity, $\|(x + y)/2\|_B = \lim_\mathcal{U} \|(x_i + y_i)/2\|_{B_i} \leq 1-\delta(\varepsilon)$.]

For another example, a space $B$ is of martingale cotype $q$ iff it is isomorphic to a Banach space with modulus of uniform convexity $\delta(\varepsilon) = C \varepsilon^q$.  In other words, there is a constant $C$ and a new norm $\| \|_0$ such that $(1/C) \|x\|_0 \leq \|x\|_B \leq C \|x\|_0$ and $(B,\|\|_0)$ is uniformly convex with modulus $\delta(\varepsilon) = C \varepsilon^q$.  Call such a $B$ a space of **martingale cotype $(C,q)$**.  Unlike martingale cotype $q$, martingale cotype $(C,q)$ is preserved by ultraproducts.  [The proof is similar to the previous one.]

**Update 2 (2015-09-27)**:  The reason I am interested in this is because such properties have a lot of uniformity associated with them.  For example, [Avigad and I][1] gave a variational norm on the mean ergodic theorem which holds for every Banach space of martingale cotype $(C,q)$.  Our inequality only depends on $C$ and $q$.

  [1]: http://arxiv.org/pdf/1203.4124v4.pdf