Nothing is impossible
Gregory Titelman's America's Popular Proverbs and Sayings says that the earliest reference to "never say never" is in Charles Dickens' first novel, The Pickwick Papers (1837). Many online sources corroborate this fact - or perhaps rather they are simply repeating it because, thanks to the wonders of online texts, we have searched The Pickwick Papers from top to bottom, and back again, without finding any reference to "never say never."
Perhaps we missed it - if we did, please do write and let us know! What we did find, however, were references to "never say die", such as:
"That's right," said Mr. Price. "Never say die. All fun, ain't it?"
At its most literal, never say die functions as a positive counter to a statement such as "we are all going to die." In other words, however hopeless a situation looks, don't give up!
Never say die is first cited in the USA in 1814 and would seem to have been in familiar use a couple of decades later when Dickens uses it; but it's believed to have hit maximum popularity in 1939 due to the movie of the same name starring Bob Hope and Martha Raye.
As for "never say never", we were unable to track down much specifically about this expression, so conclude that it is likely a variation on "never say die."
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