My friends are telling me that calling a mission peerless - a unique mission - is not correct. Is that true? I thought that unique is very much equivalent to peerless.
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You certainly can. An article in The Washington Post titled "For Searchers in Iraq, a Peerless Mission" did just that. It may not be the most common phrasing, but it is valid.
As per the comments, unique shouldn't be equated with peerless. Someone may have a unique hairstyle, but that does not make it peerless (the best you've ever seen, surpassed by none). Likewise, an actor may give a peerless performance (an unmatched rendition of Hamlet), but it may not be unique (others have done it, just not as well).
I agree with FumbleFingers that the construction sounds awkward and isn't one you'd find on a typical basis.