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Andrés's comment can be turned into an answer, but there is a slight subtlety. Suppose $M$ is a countable iterable model of ZFC + $V = L[U]$. Let $\kappa$ be its measurable cardinal and $\alpha$ be its ordinal height. Then there is a model $N\neq M$ of height $\alpha$ satisfying ZFC + $V=L[U]$ + $\kappa$ is measurable. The subtlety is that no such model can lie in $L$: since $\alpha$ and $\kappa$ are uncountable in $L$ (they're Silver indiscernibles), $N$ contains all the constructible reals, and since $N$ thinks there is a measurable cardinal, $N$ contains nonconstructible reals as well. On the other hand, let $g$ be $L$-generic for the collapse of $\alpha$ to countable. Then by $\Sigma^1_1$-absoluteness, $L[g]$ contains a model $N$ of height $\alpha$ satisfying satisfying ZFC + $V=L[U]$ + $\kappa$ is measurable. Note that $N$ is not equal to $M$ since $M$ does not belong to any generic extension of $L$: indeed $0^\#\in M$.