We say that a language $L$ is prefix-free if for every word $s\in L$ there does not exist a nonempty string $w\in\Sigma^*$ such that $sw\in L$ (i.e. no word in the language is a prefix of some other word in the language). For any given language there is a smallest prefix-free sub-language constructed by finding any two words $s,t\in L$ such that $s$ is a prefix of $t$, and removing $t$ from $L$.
Question: Is there a computable language such that its smallest prefix-free sub-language is not computable?
My attempt: I started trying to think of computable languages where "deciding" whether, for any string that is accepted, there exists a longer string which is also accepted, takes infinite time to check. None of the simple examples that come to mind work: If you take for instance any language which accepts finitely many strings, then it and its prefix-free sublanguage are computable. If you take the language of strings that ends in a 1, it and its sublanguage are computable because every string you could accept is a prefix to another accepting string so the sublanguage would be empty.
After trying to come up with simple examples I started thinking maybe I need to work in reverse, and think of a noncomputable language which is the prefix-free sublanguage of a computable language. The obvious language to pick here is the language of machine-input pairs such that the given machine halts on the given input. But I don't see how I could construe this as the prefix-free sublanguage of some other larger computable language. And I've thought a little about a few other noncomputable languages but that hasn't helped at all.