Was he actually literally gaslighted and not mentally ill at all?

The little girl was the niece of his college roommate, who was also a delusion/hallucination.
In the context of the film, I think it is very clear that he was indeed schizophrenic, in fact that’s kind of the whole point of the movie. Madness mingling alongside genius.
As for what happened in reality, I haven’t the faintest idea, other than John Nash was indeed schizophrenic (self-admitted) and learned to live with it, and he was also a brilliant mathematician. He probably wasn’t a CIA operative, or had any professional connection to the government, so why would he be gaslighted? And his work was put into use in the real world in ways that made a lot of people rich, so if anything the ‘gaslighters’ would have wanted to trick him into thinking he was sane so that he could be more productive.

John Charles Martin Nash is John Nash's son. Please, just go look into him and come back around and tell me who is naive here.

No. Synopsis: The play is set in fog-bound London in 1880, at the upper middle class home of Jack Manningham and his wife Bella. It is late afternoon, a time that Hamilton notes as the time "before the feeble dawn of gaslight and tea."

Bella is clearly on edge, and the stern reproaches of her overbearing husband (who flirts with the servants in front of his wife) make matters worse. What most perturbs Bella is Jack's unexplained disappearances from the house: he will not tell her where he is going, and this increases her anxiety. It becomes clear that Jack is intent on convincing Bella that she is going insane, even to the point of assuring her she is imagining that the gas light in the house is dimming.

The appearance of a police detective called Rough leads Bella to realise that Jack is responsible for her torment. Rough explains that the apartment above was once occupied by one Alice Barlow, a wealthy woman who was murdered for her jewels. The murderer was never found.

Jack goes to the flat each night to search for the jewels, and lighting the apartment's gas lights causes the lights to dim in the rest of the building. His footsteps in the supposedly empty apartment persuade Bella that she is "hearing things." Rough convinces Bella to assist him in exposing Jack as the murderer, which she does, but not before she takes revenge on Jack by pretending to help him escape. At the last minute she reminds him that, having gone insane, she is not accountable for her actions. The play closes with Jack being led away by the police.