Let $M$ be a model of ZF. Consider the substructure $N$ of $M$ consisting of those elements that can be well ordered. Is $N$ a model of ZFC?
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$N$ may not even be transitive, since a singleton can always be well-ordered. It's more meaningful to consider the class $HWO$ of hereditarily well-orderable sets. Even then, Exercise II.2.16 in Kunen says $HWO$ may not satisfy replacement or power set.
Actually, it seems that $HWO$ never satisfies power set axiom unless $\mathsf{AC}$ already holds in $V$. As remarked in Kunen after the above exercise, $\mathcal{P}(\kappa)\subseteq HWO$ for any $\kappa$, so if $HWO$ satisfies power set axiom we must have $\mathcal{P}(\kappa)\in HWO$, so $\mathcal{P}(\kappa)$ is well-orderable for any $\kappa$. By Kunen Exercise I.12.17, this implies $\mathsf{AC}$. It is still interesting to see when does $HWO$ satisfy replacement.
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$\begingroup$ Why is that more meaningful? Non-transitive models are still models nonetheless. $\endgroup$– Carla_Commented Aug 3 at 14:25
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4$\begingroup$ @Carla_ Most of the natural models studied in set theory are transitive. A random non-transitive model $(M,\in)$ may not even satisfy extensionality. Indeed, as long as there exists some non-well-orderable set $x$, we have $\{x\}\in WO$, so extensionality fails in $WO$ because $\{x\}$ and $\emptyset$ have the same elements but are different sets. $\endgroup$ Commented Aug 3 at 14:33