This sentence has been occupying me for a while now, since I guess it shows there is indeed no linear match between all the possible tenses in English and Italian, with no direct equivalent here existing for the English present imperfect continuous tense.
Without any context I would interpret the Italian sentence as “I am living in a cave”, and actually, I guess there’s nothing really wrong with that, since the present imperfect continuous tense used in the English sentence also shows that it is still a currently ongoing activity/state, so it is still true, and the person who is currently living in the cave will have been living there for a while too, since you’re talking to them now, and they’re not saying “I’m going to be living in a cave”, “I have just moved into a cave”, or “I am in the process of moving into a cave”.
Still, it’s been bugging me a little for some strange reason. [Mi sta preoccupando ;-)]
At least, in addition to the very helpful easitalian website describing the various Italian tenses I had already discovered before, it lead to the happy discovery of the Collins Grammar Dictionary being easily accessible online too.
The tenses do not match completely, but there is no way to convey the exact meaning of the English sentence in Italian by verb conjugation alone. At least, none that I can think of at the moment.
You can qualify the Italian sentence like “È da un po’ che sto vivendo in una caverna” (or even “che vivo in una caverna”, the indicative present can be used for ongoing actions too). This is a way to convey that the action ongoing now started some time ago.
“Ho vissuto in una caverna” suggests that I am not currently doing so.