From my book,
However, Jisho says that 遅く can mean 'slowly,' but I can't find an example for that meaning in the same source.
In this thread, answerers agree that 遅く can be used to mean 'slowly.'
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遅く can mean slowly, but ゆっくり is almost always preferred in non-technical contexts. もっと遅く食べた方がいいよ and 彼は遅く走る are not completely wrong but do sound fairly unnatural to me.
遅く meaning slowly may be naturally used in scientific contexts because ゆっくり can have unwanted nuances like "relaxed; leisurely". For example, saying 物体が光の速度に近づくと時間が遅く流れる referring to the theory of relativity looks natural to me.