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The quote was taken from, "The River War", by W.L.S. Churchill. You have to watch what Manchester says, but he got this one right. (I don't know where Manchester said the quote came from; but that's where Churchill originally mentioned it.) The battle was at Omdurman, in the Sudan.
He may have also mentioned it in, "My Early Life", which I haven't read for some years. It was re-released about the time that, "Young Winston" was filmed, and libraries may have copies. Very witty book, too.
Manchester has made some anti-gun comments. I'm mildly surprised that he brought up Churchill's pistol.
Churchill had several other handguns, too. He especially liked a Colt M-1911 that he bought for WWI, and one famous photo of him in a suit in North Africa or at Yalta shows the outline of the gun. His autobiography notes that when WW II began, "I got out my pistols and armed myself."
Lone Star