Bdtyk

Can anyone nail down the origin and first usage of the phrase...

“barking mad”
(also, just "barking")?

Basic research shows two possible answers:

Barking, England supposedly had a mental institution attached to the abbey back in the medieval period.

However, according to the author of one post on -phrasefinder.org

”The problem with the asylum tale is the date - it is far too early. 'Barking mad' isn't medieval and began to appear in the language only around the beginning of the 20th century
The first record of it that I can find in print is from the USA. The 11th November 1927 edition of the Oklahoma newspaper The Ada Evening News….”

This would seem to be incorrect.
I have found a number of passages from the 19th century going back as far as 1826, mostly from BrE texts.

As well, I have only seen it in BrE usage such as in the Harry Potter series:

"Barking," said Uncle Vernon, "howling mad, the lot of them. You'll see. You just wait."

"Barking mad"
- Ron on Hermione

"It's not my fault she's barking mad"
- Harry referring to Madam Pince

Also, there is the obvious reference to a “mad dog”, as seen in the 1826 text.

“A very respectable paper in this county, too, only three years ago, published an account of a horse bitten by a presumed mad-dog, going barking mad. Who, after this, can depend on newspaper records of Hydrophobia ”

Is there any source lurking out there behind a pay-wall or other print media that could indicate the origin of this phrase?

As a supplement to lbf's interesting and useful answer, I note one surprisingly early relevant instance of the phrase "barking mad." From a letter "To the Printer of the London Chronicle" by Dr. Henry Bracken, published in the London Chronicle (October 23–25, 1760):

Thus far I know, viz. that, about 14 years ago, a man near Garftang, in this county, happening to go into his barn, and finding a strange dog, he took up a pitch-fork to drive it out, and the dog leaping up at the man, bit him into his lip, and then made away; so a messenger was immediately dispatched to Mr. Hill, and the specific powder was given without effect; for the poor man died barking mad in a very little time : and I could name other instances in brutes, as cows and dogs, where this cried-up specific has failed of success. But indeed every county in England has one or more, who pretend to sell something to others as a certain cure for the bite of a mad dog : and not only England, but in most other kingdoms, it is the same with respect to these empirical proceedings.

I couldn't find a copy of this periodical that provides a full view of its contents, but stalwart site participant JEL did—and I have altered my link accordingly.

I did find an excerpt from Dr. Harken's letter (including the "barking mad" passage) in William Robertson & Caleb Parry, Tentamen Medicum Inauguralis de Rabie Contagiosa (1778). A full version of the letter is to be found in Joseph Dalby, The Virtues of Cinnabar and Musk, Against the Bite of a Mad Dog (1762). Dalby and Bracken were evidently bitter enemies in the rabies cure racket, with Dalby championing a preparation of cinnabar and musk, while Bracken favored a nostrum that Dalby dismissed as "goose-grease." Clearly the best course was not to get rabies.

It thus appears that use of the phrase "barking mad" in reference to someone infected with (and dying of) hydrophobia after being bitten by a mad dog goes back at least to 1760—a time when the nature of the disease was obviously not well understood.

Dictionaries and Etymonline are lacking. But barking mad from World Wide Words barking mad seems to have it right.

Nicholas Shearing of the OED kindly hunted through their database of
citations and found that their earliest reference is actually from as
far back as 1933, from Mr Jiggins of Jigginstown by Christine Pakenham
(Countess Longford): “But he was mad! Barking mad!”.

But these via @Sven Yargs dates it to 1870 and 1853: Twelve Wonderful Tales Told in Rhyme - google books and another google book entry and 1851 google books.

reference 2:

"My tale is nearly ended, and there's little more to add,/ Except that
in a week or so the Count went barking mad,/ And took to biting
people's legs, a joke extremely bad ..."

It's a long shot, but there might be a connection with Berkshire Hunt, Cockney rhyming slang for cunt. Berkshire is pronounced Barkshire in UK. Hunt = dogs, who tend to get a bit agitated when they have chased a fox up a tree, and sometimes have to be dragged off, perhaps foaming at the mouth. I don't claim to be an expert on the barbaric spectacle of fox hunting, being of like mind to Oscar Wilde: the unspeakable in pursuit of the uneatable.