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to become; to appear like; to have a touch of; to look like
「染みる」 has another reading with different meanings.
年寄り染みた話し方が気になった。
I noticed his way of speaking that seemed old-mannish.
彼の世間慣れした態度が大人染みていた。
His worldly-wise attitude seemed very adult.
子供染みた振る舞いはやめてほしい。
I want you to stop the childish behavior.