There is an American TV series called “Mad Men”, about advertising executives in 1960s New York. The title is a pun on “Madison Avenue” and “Advertising” (Ad Men). In US English, “mad” means “angry”, whereas in the UK it only means “insane”. So to British ears it’s a different sort of pun.
In my own speech (South-east England) it’s also a minimal pair with either a short and long vowel. To make it mean “insane” I have to lengthen the vowel in “mad”. For the “Madison/Advertising” meaning I have to shorten it, so the pun is wrong on two counts for me.
Most varieties of English have a long /a/ as in “father” and a short one as in “dad”. Southern British speakers use the long vowel in words like path, bath, past, where most Americans and Northern English speakers use the short one. However, some speakers like me from Southern England have a third phoneme, with the same vowel quality as in dad, add, crack, but about twice as long. It only occurs before some voiced consonants. So I have, for example:
Short vowel, like everybody else: dad, pad, sat, bang, crack, gang
Long vowel, unlike most speakers: sad, jam, bad, mad (insane)
Phoneticians would call these two vowels “allophones” of the same phoneme (variants which depend purely on the positional context of the sound) unless there were at least one minimal pair, in which two words mean different things but are contrasted only by this sound. I have only ever found 2 such minimal pairs in my own speech. One is “mad”, as described above, and the other is “jam”, where I use the short vowel in the context of “traffic jam” but the long one for “strawberry jam”.