(The synonymous verbs besmirch or sully are much more
common.)
The accusations did not seriously smirch the author's
reputation.
Walking through dense foliage can smirch or stain light-colored
clothing.
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The word you are looking for is "slander." Slander refers to making false and damaging statements about someone with the intention to harm their reputation.
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besmirch
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In a final attempt to defame the family the reporter posted a nasty article.