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Here's an argument for an affirmative answer to Joel's modified version of the question. Suppose we have a forcing that preserves cardinals but singularizes some cardinal $\lambda$ that was regular in the ground model. Note that $\lambda$ had to be a limit cardinal, since otherwise singularizing it would collapse it down to its immediate predecessor cardinal (if not even lower). Now let $C$ in the forcing extension be a cofinal subset of $\lambda$ of smaller cardinality $\kappa$. I claim that C is not included in any set $D$ in the ground model of cardinality $\leq\max\{\kappa,\aleph_1\}$; in other words, I claim that $C$ is a counterexample to the assertion that the forcing extension satisfies the covering lemma over the ground model. Indeed, suppose we had such a $D$. Intersecting it with $\lambda$, we'd have a cofinal subset of $\lambda$ strictly smaller than $\lambda$ in the ground model, contrary to the assumption that $\lambda$ is regular in the ground model. ("Strictly smaller" in the preceding sentence uses that $\lambda>\aleph_1$, which is why I pointed out earlier that $\lambda$ has to be a limit cardinal.) So the forcing extension doesn't satisfy the covering lemma over the ground model. That implies the existence of an inner model with a measurable cardinal, by an ancient result of mine --- "Small extensions of models of set theory" in "Axiomatic Set Theory" (Proc. of 1983 Boulder Conference, edited by Baumgartner, Martin, and Shelah) Contemporary Math. 31 (1984) pp. 35-39.