Faux-naïf (faux-naif) is a French loanword which literally means falsely naive. In English its meaning is much the same, a person or object pretending to be innocent or unknowing in order to fool someone or get something underhandedly. Faux-naif can be an adjective or a noun.
It can be spelled with or without the ï, though it is listed in the dictionary as with. The two versions most likely comes from the assimilation into English. Its first known use was relatively recent, 1948, and we suspect the regular i version will overtake the other if given enough time.
In another instance of differences between British and American English, the British version is listed without the hyphen unless used as an adjective before the object it describes.
In American English faux-naif is listed with a hyphen in most dictionaries, but some say it is okay to use it without the hyphen. Others list the noun version without a hyphen. We think this is decidedly a word in transition, as are most hyphenated words (and loanwords). When in doubt, use a hyphen.
Examples
An ensemble of lovable misfits – adults costumed as and behaving much like schoolkids – enact rituals of the absurd to largely nonsensical, faux naïf songs. [Huffington Post]
It may be the clown in him, but Thierrée seems unable to allow any sequence to play out without introducing some cute or faux-naïf piece of physical business. [The Guardian]
While Avati has always preferred faux-naif as a style for his sometimes deceptively simple stories, this film has a different, more up-to-date look. [Hollywood Reporter]
There will always be odious half-wits in this country, as elsewhere, and it would be faux naif to affect shock about such a poisonously demented response to a proud, if expatriate, Scot expressing his opinion about his idea of the best future available to his country. [The Telegraph]