The ghost was identified as Major William Henry Braddell, who had been a member until his death. Legend has it that, on his last visit to the club in May 1940, he had been set to dine with two friends, and left the room fleetingly, to return and find it bombed out, and his friends dead. Around a week later, the Major himself was killed in an air raid. It is thought he came back to haunt the club because he felt so at home there.
According to Steve Roud in London Lore, which your author received for Christmas, when The Sun covered the story in 1994, it reported that the club would not be calling in an exorcist, as Major Braddell was still a member.
For more on the story, see here.
***Edit - A reader more knowledgable than your author points out that the ghost was at the club's former premises (pictured) in Cambridge House, Piccadilly, and the club purchased a new premises at St James's Square in 1996, to which it moved in 1999. The ghost was, therefore, at the club's old premises. Sorry!***