As I've noted in a previous question, the definition of an extender seemingly makes it impossible to ensure that the ultrapower by an extender is closed under countable subsets. That's inconvenient because it would mean that no mainstream ultrapower construction can combine the properties $V_\gamma \subset M$ and $M^{\lt \lambda} \subset M$ for $\omega \le \lambda \lt cf(\beth_\gamma)$, like in the definition of ultrahuge cardinals.
But is the extender construction incapable of ensuring closure properties? Last time I asked the question I got an answer that pointed out obvious obstructions in certain cases, but no answer to the general question. Thus I change my question to rule out those obstructions: Suppose that $\mathcal{E}$ is a $(\kappa, \gamma)$-extender (or a $(\kappa, \beth_{\gamma}+1)$-extender; I suspect that the definition varies) derived from an elementary embedding $j: V \to M$ such that $V_\gamma \subset M$ and $j_{\mathcal{E}}: V \to M_{\mathcal{E}}$ is the ultrapower embedding by $\mathcal{E}$. Is $M_{\mathcal{E}}$ closed under subsets of cardinality $\lt min (\kappa^+, cf (\beth_\lambda)$?