The horrible thing about the Two Minutes Hate was not that one was obliged to act a part, but, on the contrary, that it was impossible to avoid joining in. [...]. And yet the rage that one felt was an abstract, undirected emotion which could be switched from one object to another like the flame of a blowlamp. Thus, at one moment Winston's hatred was not turned against Goldstein at all, but, on the contrary, against Big Brother, the Party, and the Thought Police; [...]. And yet the very next instant he was at one with the people about him, and all that was said of Goldstein seemed to him to be true. At those moments his secret loathing of Big Brother changed into adoration, and Big Brother seemed to tower up, an invincible, fearless protector, standing like a rock against the hordes of Asia[...]
It is clear that the hate speech is canalizing the accumulated aggression of the individuals and in this case both Big Brother and Goldstein are mere projections.
There were occasions when Big Brother devoted his Order for the Day to commemorating some humble, rank-and-file Party member whose life and death he held up as an example worthy to be followed. Today he should commemorate Comrade Ogilvy. It was true that there was no such person as Comrade Ogilvy, but a few lines of print and a couple of faked photographs would soon bring him into existence.
So again and again we get confirmation that persons are not only "vanished" in tradition of damnatio memoriae, but also invented for purposes from the Party.
But in the same moment, drawing a deep sigh of relief from everybody, the hostile figure melted into the face of Big Brother, black-haired, black-moustachio'd, full of power and mysterious calm, and so vast that it almost filled up the screen.
This is the only physical description we ever get from the book. Later this hypnotic gaze is added, but it seems that is everything.
At the apex of the pyramid comes Big Brother. Big Brother is infallible and all-powerful. Every success, every achievement, every victory, every scientific discovery, all knowledge, all wisdom, all happiness, all virtue, are held to issue directly from his leadership and inspiration. Nobody has ever seen Big Brother. He is a face on the hoardings, a voice on the telescreen. We may be reasonably sure that he will never die, and there is already considerable uncertainty as to when he was born. Big Brother is the guise in which the Party chooses to exhibit itself to the world. His function is to act as a focusing point for love, fear, and reverence, emotions which are more easily felt towards an individual than towards an organization.
So we have in fact two concepts which must be separated.
The first concept is "the" Big Brother which is the image portrayed by the Party. As already said in the last sentence his function as party leader is to give the party a venerable face.
This Big Brother cannot exist for someone which is not indoctrinated as real person because he is flawless and therefore impossible (I added "not indoctrinated" because there are unfortunately real life examples of extreme veneration).
We can answer the question if the Party Big Brother exists with a unambigous No.
But the question remains if there is a real person behind this image. It is quite normal that behind a facade of hyperboles an individual as party leader exist. So does a real Big Brother exist ?
The past, he reflected, had not merely been altered, it had been actually destroyed. For how could you establish even the most obvious fact when there existed no record outside your own memory? He tried to remember in what year he had first heard mention of Big Brother. He thought it must have been at some time in the sixties, but it was impossible to be certain. In the Party histories, of course, Big Brother figured as the leader and guardian of the Revolution since its very earliest days. His exploits had been gradually pushed backwards in time until already they extended into the fabulous world of the forties and the thirties, when the capitalists in their strange cylindrical hats still rode through the streets of London in great gleaming motor-cars or horse carriages with glass sides. There was no knowing how much of this legend was true and how much invented. Winston could not even remember at what date the Party itself had come into existence. He did not believe he had ever heard the word Ingsoc before 1960, but it was possible that in its Oldspeak form-'English Socialism', that is to say -- it had been current earlier. Everything melted into mist. Sometimes, indeed, you could put your finger on a definite lie. It was not true, for example, as was claimed in the Party history books, that the Party had invented aeroplanes. He remembered aeroplanes since his earliest childhood. But you could prove nothing. There was never any evidence.
From that we can safely conclude that there may have been a person, probably the leader of INGSOC, which was known as Big Brother; but as the whole history and every evidence is permanently destroyed and legendaries, venerations and feats added, we can also safely conclude that an actual person would have been completely different from the Big Brother presented. It is even possible that INGSOC started with an idolization of a person, the perfect leader, and that there never was a Big Brother.
A person which may resemble the Big Brother marginally may have existed, but it never could achieve the venaration and adulation the party desires because a person can be met and therefore their impression can be tarnished, they can be attacked and killed by political opponents, they could change their mind against their party or simply die.
For this we can conclude from the cited passages that we have not the easy option of a canon answer because the novel (deliberately) cannot give an answer: The narrator has only access to Winston's knowledge who is living in a world of pathological untruth.
We do not know.
Winston himself wants to break his ignorance about Big Brother's realness:
Winston asks O'Brien:
'Does Big Brother exist?'
'Of course he exists. The Party exists. Big Brother is the embodiment of the Party.'
'Does he exist in the same way as I exist?
'You do not exist,' said O'Brien.[...]
'I think I exist,' he said wearily. 'I am conscious of my own identity. I was born and I shall die. I have arms and legs. I occupy a particular point in space. No other solid object can occupy the same point simultaneously. In that sense, does Big Brother exist?'
'It is of no importance. He exists.'
'Will Big Brother ever die?'
'Of course not. How could he die? Next question.'
This dialogue offers a much more sinister and disturbing viewpoint. The novel is not only about a failed uprising in a dictatorship. It is that this system is so rotten that nothing, really nothing can be relied on, not even Winston's writings or the storyteller who depends on Winston's experiences. The system destroys every evidence someone can count on, it denies the inviolable acknowledgement of a person and it denies you therefore a consistent image of your self and of the world.
Winston MUST NOT KNOW if Big Brother exist. Winston HAS NO RIGHT TO EXIST.
In a setting like this there can be no Big Brother or two-hundred Big Brothers because what you call "reality" is denied by definition.
Therefore the most famous scene: 2 + 2 = 5.
Given the circumstances we cannot correctly answer the question because every evidence in the story is tampered with, even the emotions and the memories of the participants. Perhaps the article gaslighting gives a small impression how traumatizing this environment is. This environment is capable to do that 2 + 2 = 5 and I think this is the message Orwell wanted to deliver.
Let's imagine after the end of the novel Orwell adds after the next page: "P.S. The Big Brother exists/does not exist/is in fact an order of Nine Chosen Ones ?"
We can ask what impact would that have on the novel ?
It is a break with the narrator structure. In the whole novel we can only experience what Winston experiences, thinks or remembers, we have no direct insight about O'Brien or Julia.
So we as reader are exactly as clueless as Winston, we experience Winston powerlessness and we are denied answers like Winston. Worse, we experience the reality bending that even the concept of truth or the acknowledgement of our existence is denied. As the whole point of the novel is to show us what the System do to Winston, the fictional postscriptum from Orwell does not belong to the story because it presents additional facts Winston cannot know, may not know and must not know. Therefore it cannot be a part of the story and it really does not matter at all if Big Brother is a real person or not.