Berkeley's immaterialist arguments are quite interesting. And since philosophy does not defer entirely to experimental "proof," metaphysical arguments can't be flat out "proven" or "disproven" in that sense. One must examine the coherence and assumptions in the argument.
First, Berkeley is not arguing for a Matrix-type skepticism. He would argue that sensory experience of objects is perfectly "real," only it is not caused by something "behind" or "below" our experiences in the sense of Kant's noumena or Locke's primary qualities. That includes some inert underlying substance we cannot perceive named "matter" in Newtonian mechanics.
So Berkeley is a very strict empiricist. All we have are "experiences" and our recombinant "ideas" about experiences, such as memories, imagined things, which are all like shakier, fainter experiences. All of these he calls our "ideas," only some are very vivid and immediately sensed. It makes no epistemological sense, he argues, to discuss things like "matter" that could on principle never be experienced. Though, of course, we can make up signs and names as matters of convenience.
The most famous refutation of Berkeley's idealist empiricism is Samuel Johnson's, who said "I refute it thus!" and kicked a stone. This appeals to the "common sense" idea that there is a vividly distinct ontological class of mind independent objects, an argument which can also be demonstrated in a less terse, comical fashion.
But again, Berkeley is not to be misunderstood as a skeptic. Experiences are real. Empirical science is valid. His primary aim amounts to a novel proof of God by a careful examination of materialist, mechanical doctrines, particularly causation. Like Hume and Hegel after him, he believes material causation is simply "constant conjunction" observed by a mind. It is not something essential and universal beyond observation.
For Berkeley, only minds are active, creative, and "cause" things. We know this for certain because we, as finite minds or spirits, can immediately create or "cause" new ideas and new experiences. Though we cannot seem to cause those "ideas of sense" or what we would call "objects." Thus, he believes, we must account for those things my some other mind, since mind is the only creative or "causal" force we immediately experience. Causation or creation cannot arise from something presumably inert and never even experienced, called "matter."
So we get the same "common sense" world, only mind comes prior to matter, which creates nothing but is itself created by mind, not our own finite minds but infinite mind. Why, Berkeley might argue, do we insist on imagining an infinite, dead, inanimate universe preceding and surrounding minds, which somehow arose out of it? This chicken-egg conundrum is contrary to what we in fact know most immediately and vividly, which is experiences and a mind unifying them. We have no "experience" of unconsciousness prior to or distinct from consciousness. How could we?
So, rather than rehearse "mind independent" arguments against Berkeley, of which there are many, I'm just pointing out that he is not a skeptic, not saying that our experiences are illusions or that there is something "behind" the matrix of perceptions "causing" our delusory ideas. His arguments are not meant to raise doubts, but to relieve us of the mistaken abstraction that we are inexplicable sparks of mentality fallen into a vast, dead, mindless universe that immediately extinguishes us. He might wave his hands at all that we are experiencing now and reply to Johnson "I refute it thus!"