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Explanation: The idea combines work over time, constancy (even stubbornness), and an objective - maybe getting the Christmas lunch washing-up done, maybe getting the thesis in on time - and corresponds to the feeling of English phrasal verbs using "away".
No idea how to explain how the words come to mean what they mean here, as with a lot of slang.
But the translation has to echo the flavour of the original, and "non-stop" is not very slangy - it appears in airline/train timetables, so I think we've got to look for something else. Hence "slogging away", "banging away", even "working away"; what about "(working) my guts out"?
Any ideas anyone? (I've got the post-Christmas cold - un trancazo de tres pares - and can't think, but I'm working on it.)