I think a better way to unsplit the infinitive while (definitely) keeping the meaning is, "I do not have time to be always studying." This keeps the "always" with the "studying", as Vlada suggests. Though it may sound weirder to some, and I don't know if it'd be accepted. As you wrote it, it seems unclear what "always" is modifying -- and I can imagine both possibilities in different spoken phrasings. Exactly how/if the meaning changes depending on the "always" is another fun question.
I would prefer "Nemám vždy čas se učit.". Other forms are possible, but the risk of confusion is large. "Nemám stále čas se učit." could mean that as well but can also mean you do not have the time yet. Your proposed order might be acceptable but does not sound too well.