Too smart for his/her/its own good (British clever being roughly equivalent to U.S. smart). OED notes the first use of … by half formulation in Sheridan’s 1780 School for Scandal–“Oh, he’s too moral by half”–and of too clever by half in 1858. As the Punch cartoon suggests, it quickly became a popular catchphrase and remains so.
Google Ngram shows an interesting (and common) pattern in U.S. and Britain. British use peaks in about 1980, then declines. American use has roughly doubled since 1950. Currently, use is about equal in both countries.
Some Republican leaders, like their Democratic counterparts, find [Richard] Darman too clever by half. (Elizabeth Drew, the New Yorker, April 9, 1990)/Everything in Hartford seems too clever by half. (TheDay.com [New London, Ct.], June 12, 2011)
notoneoffbritishisms.com 
In British English, “smart” is more often used of appearance than intelligence, although it can refer to either.
Interesting. I didn’t realize that British ever used “smart” for intelligent. In U.S., “clever” is often (usually?) a backhanded compliment, with the word “merely” assumed before it.
The British certainly use ‘smart’ with the American meaning, but it’s usually mildly derogatory, for example to refer to someone trying to look clever and coming across and smug or superior. A common phrase, certainly in the Home Counties, is ‘smart-arse’, which certainly has negative connotations in this way. I was often told growing up “no-one likes a smart-arse”!
With regards to “too clever by half”, I suspect there are many variations on this – again I can only really speak for the south-east of England, but the version we used was “so sharp you’ll cut yourself”, and I’m sure there are plenty more!