Origin of the term 'K-9 unit'
An Elephind newspaper database search turns up at least one instance of "K-9 unit" from World War II. From "5th Infantry Dog Up to Snuff With '201' File" in the [Colorado Springs, Colorado] Camp Carson Mountaineer (November 11, 1943):
It is not unusual for a soldier to have a personal (201) file in Regimental Headquarters. This is standard operating procedure. Not is it news when a dog bites a man. Bu this is a case where the dog involved did not bite anyone—but has a file (201) all his own.
The canine in question answers to the name of Charley. He is owned and operated by the Regimental Adjutant, Capt. Adelbert Boggs. Charley i a fine, year-old Doberman Pinscher. He joined the Regiment way down yonder at Camp Van Dorn.
The file keeper, S/Sgt. Saul Kash is in charge of Charley's 201 file as well as those of the entire Regiment. He says that Charley's dossier is quite complete. It contains Charley's admittance date to the Fifth—and very valuable documents surrounding the pup's efforts to join the fighting K-9 unit. This, as you know is the corps of trained dogs which is proving of such invaluable assistance to our sentries everywhere.
Newspaper references to "K-9 corps" during the same year are far more numerous, however. Here are the earliest three such instances. From "Two Coronado Dogs Accepted by Army's 'Dogs for Defense'," in the Coronado [California] Journal (January 7. 1943):
Forty-one San Diego county dogs to date have been accepted for service in the new K-9 corps of the United States Army, it was announced by officials of Dogs for Denfense, Inc., in charge of recruiting of the canine-soldiers in San Diego and Imperial counties.
"Denfense" appears to be a typo, but with stories like this one, you never know.
From "Falla's a Private," in the [Newburgh, New York] Stewart Field United States Military Academy Prop Wash (March 9, 1943):
Falla, President Roosevelt’s Scottish terrier, has gone to war. He hasn’t joined the K-9 Corps, for he isn’t big enough. But he’s a private, thanks to the President’s personal $1 donation to the War Dog Fund. The dollar entitled Falla to the rank of either an Army private or a Navy seaman, and the President asked that Falla be made a private. War Dog Fund has turned over $15,000 so far to Dogs for Defense, national procurement agency of dogs being used by the various military services at home and abroad.
From an untitled item in the Suttons Bay [Michigan] Courier (April 8, 1943):
They’re in the army now—bow wow, and how! Tens of thousands of America’s dags have gone to war, in the K-9 corps. Read of the splendid job they’re doing and how they’re trained, as told by Louis de Casanova, former editor of the Kennel Club Gazette, in The American Weekly, the magazine distributed with next week's Chicago Herald-American.
None of these early instances explains the origin of the name—but presumably that's because it was obviously chosen as a visual pun—making the verbal connection between the word canine and the military inclination to organize individuals (whether people or dogs) into groups with numerical IDs. The fact that "K-9 corps" and "K-9 unit" were appearing in newspapers across the country—from New York to Michigan to Colorado to California (along with instances in newspapers from Indiana, Texas, Virginia, and Hawaii)—within the year 1943 indicates that the name caught the fancy of working journalists (and evidently the U.S. public as well). I didn't find any mentions of "K-9 unit" or "K-9 corps" in the Elephind database from 1942 or earlier.
Man-dog team or police department subunit?
It seems worth noting that "K-9 unit"—like "K-9 corps"—originally applied to a section or group of trained military dogs, not to a particular pairing of a dog and its handler. In modern policing, however, people sometimes apply the the term "K-9 unit" to a man-dog team and sometimes to the unit of the police force that consists of such teams. Thus, for example, from "Local Man Fires Shot at Burglars, Suspects Escape Police Dragnet," in the Louisville, Colorado Times (December 5, 1990, page 11) we have an instance of the "man-dag team" sense of the term:
Louisville police also requested a K-9 unit from the Boulder County Sheriffs Office, but no dog units were available for the search. Somehow the two young burglars managed to elude the dragnet. Glass said their escape was especially frustrating, as they may be linked to burglaries in Broomfield and Longmont.
But far more common in Elephind's newspaper matches is the term in its departmental sense, as in this instance from Shawn Dockry, "Kent Police Praise Dog Unit," in the [Kent, Ohio] Daily Kent Stater (April 5, 1991):
The Kent City Police and the Portage County Sheriffs departments literally want to take a bite out of crime with their respective K-9 units.
Portage County Sheriff P. Ken Howe said his dog unit started six months ago with one dog, Erd, pronounced 'Ed.' A new dog was added to the unit in March.
"The dogs are to us what the horses are to Cleveland (police units)," Howe said.
The creation of the K-9 unit was a campaign promise of 1988, Howe said. The dogs are used to track lost people and suspects, to sniff out drugs and to back up officers.