The colloquial American-English phrase as busy as, or busier than, a one-armed paperhanger, and variants, mean exceedingly busy.
—Cf. also ‘as busy as a one-armed taxi-driver with crabs’ and other colourful phrases.

Remark: An earlier British-English phrase, as busy as the devil in a high wind, was recorded by the English antiquary and lexicographer Francis Grose (1731-1791) in A Classical Dictionary of the Vulgar Tongue (London: Printed for S. Hooper, 1788):

Busy. As busy as the devil in a high wind; as busy as a hen with one chick.

The English author and political reformer William Cobbett (1763-1835) used this phrase in To the People of Dover. On the triumph of the Queen and the People over persecution, carried on by the means of conspiracy, subornation and perjury, published in Cobbett’s Weekly Political Register (London, England) of Saturday 18^th November 1820:

In the mean while the Queen’s law advisers are, I dare say, as busy as the Devil in a high wind.

These are, in chronological order, the earliest occurrences that I have found of the phrase as busy as, or busier than, a one-armed paperhanger, and variants:

1-: From the following advertisement, published in The Pacific Commercial Advertiser (Honolulu, Hawaii) of Tuesday 19^th June 1906:

THE FAVORITE GROTTO.

Our chef is kept as busy as a one-armed man engaged in the exercise of bean-bag. The reason is easily found. Today’s lunch menu, which follows, explains it:

SOUP—SCOTCH BROTH, BARLEY
CURRIED LAMB A LA BOMBAY,
BOILED RICE
GERMAN POT ROAST, POTATO
PANCAKE
MASHED POTATO      SPINACH
METROPOLE SALAD
25c.
With Beer, Wine, Tea or Coffee.
Open all night.
F. J. KILEY, Prop.,
Cor. Hotel and Bethel Streets.

2-: The phrase as busy as a one-armed man engaged in the exercise of bean-bag occurs again in an advertisement for the same restaurant, published the following day in The Pacific Commercial Advertiser.

3-: From The Ethics of Pig, by the U.S. short-story writer O. Henry (William Sydney Porter – 1862-1910), published in Munsey’s Magazine (New York—London: The Frank A. Munsey Company, Publishers) of October 1906:

I got as busy as a one-armed man with the nettle-rash pasting on wall-paper.

4-: From Sunday Prizefight in Little Old New York. The Kind of “Sport” Indulged in Because Country Legislators Will Not Allow Legitimate Boxing in the Big Cities, by ‘Tad’, published in The Buffalo Enquirer (Buffalo, New York) of Monday 15^th October 1906:

New York, Oct. 15.—Jeff O’Connell, an English featherweight, fought Charley Lucas, a New York lad, yesterday afternoon on a west side pier and battered him up in a fifteen round fight.
It was at 3 o’clock when the meeting was to take place. We all met in a bum saloon over on 8^th Avanue [sic], somewhere near 37^th Street, and waited for the mitt artists. There was a restaurant upstairs. A man with red grogans was the chef and he was as busy as a one-armed paper hanger with the hives.

5-: From Novices at Kicking. Americans Are Not the Champion Grumblers. Europeans Are Experts, by Clarence L. Cullen, published in The Evening Star (Washington, D.C.) of Saturday 20^th October 1906—the author described:

a Russian grand duke, a young man said to be worth some few hundreds of millions of dollars, a spender from away back, a man who on his tour of this country was as busy as a one-armed paper-hanger with the hives just tossing away money.

6-: From Letters From An Old Sport To His Son In College, by Jim Nasium, published in the Chattanooga Daily Times (Chattanooga, Tennessee) of Sunday 28^th October 1906:

Dear Son—The bats in your garret must be flapping their wings again. If you hoop up with that gang of pikers to play professional football it’s bye-bye to the knowledge factory and back to the wheat belt for you. There’s something better ahead of you, and you just stay right where you are and keep busier than a one-armed philosopher with the itch working in a high wind.

7-: From Split Finney Wonders Why They Do It. Will Some One Please Explain Why Men With Money Try to Break Their Necks?, published in The Sun (New York City, N.Y.) of Sunday 4^th November 1906:

“Bo, as long as I’ve got a two piastre shinplaster in the kick or in sight I’m going to be as busy ducking trouble as a one-armed paperhanger with the hives.”

8-: From Letters From an Old Sport To His Son in College, by Jim Nasium, published in the Chattanooga Daily Times (Chattanooga, Tennessee) of Sunday 28^th April 1907:

If you would succeed in baseball when you get on the field you want to keep busier than a one-armed paper-hanger with the hives.

9-: From Letters From An Old Sport To His Son In College, by Jim Nasium, published in the Chattanooga Daily Times (Chattanooga, Tennessee) of Sunday 16^th June 1907:

There are no sleeping cars on the limited express to fame; it’s a work train from one end of the line to the other. The sooner you get wise to this the sooner you’ll get out of your slump and into the sporting page.
So you stick right on the job, son, and keep busier than a one-armed bill-poster with the itch, working in a high wind. When you’re in baseball as a business you’ve got to treat it like a business and not go at it like yyou [sic] would a game of drop the hankerchief [sic] at a Sunday school picnic.

10-: From a letter to the Editor, published in The Chattanooga News (Chattanooga, Tennessee) of Saturday 14^th September 1907:

Mr. Hale and Sanders have been as busy as a one-armed bill poster on a wintry day with the seven-year itch.