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The following definition of the Australian-English expression drop bear is from Australia’s funniest yarns: A humorous collection of colourful yarns and true tales from life on the land (Crows Nest (New South Wales): Allen & Unwin, 2019), by the Australian scholar Graham Seal [chapter 1: Bull; page unnumbered]:

Drop bears are mythical creatures of Australian tall-tale tradition that fall from the trees onto unsuspecting dupes walking below. They are often described as koalas with large heads and sharp teeth. Like a good portion of Australian folktales, drop bears are more of a floating motif around which brief narratives can be constructed by the teller for the edification or, in these cases, the trepidation, of audiences, rather than fully formed, elaborate tales.

The Australian Museum (Sydney, New South Wales) perpetuates this tall-tale tradition, with a page on its website written as if drop bears were real—extract:

Danger to humans
[…]
There are some suggested folk remedies that are said to act as a repellent to Drop Bears, these include having forks in the hair or Vegemite or toothpaste spread behind the ears. There is no evidence to suggest that any such repellents work.

The expression drop bear occurs, for example, in the following from Yankee Sex Guru in Biochemical Popcorn Explosion: Trigger Finger Amputated by Mal Meninga, published in Woroni (Canberra, Australian Capital Territory, Australia) of Monday 21^st October 1991 [page unnumbered, column 1]—Woroni is the student newspaper of the Australian National University:
—Context: “Fresh from the agricultural backwaters of the University of California at Davis, Burton and Garran Hall’s favourite half-deaf Ugly American exchange student, Phil Holsten, takes a look at life in Australia over the past eight months”:

The vast majority of stereotypes that Australians have for Americans (and not just the ones derived from the Phil Donahue Show) are actually true. Just like in Australia where kangaroos hop down the urban center streets, where there’s no indoor plumbing, and where drop bears and bush chooks are the terrorists of the bush, normally while in America I carry a loaded AK-47 sub machine gun and have sued my parents for taking away my Porsche 944 (my only reasonable means of transportation to my uni where my fees are $500,000 a year).

The earliest occurrences of drop bear that I have found indicate that this expression originated among Australian soldiers. It was certainly well established in 1967 [cf., below, quotation 1], but the date and circumstances of its coinage are unknown. However, according to the Australian author Nancy Keesing (1923-1993) in Lily on the Dustbin: Slang of Australian Women and Families (Penguin Books Australia Ltd., 1982) [page 85]:

The ‘drop bears’ […] were invented during World War II for the benefit of gullible American servicemen.

The earliest occurrences of the expression drop bear that I have found are as follows, in chronological order:

1-: From Australian Army: The Soldiers’ Newspaper (Canberra, Australian Capital Territory, Australia) of Thursday 12^th October 1967 [page 1, columns 2 & 3]:
—The expression drop bear was already well established, since the author did not feel the need to explain it, and even joked about it (the issue has only twelve pages, so the reference to page 13 is humorous):

Realism, a bonus and a drop bear

Exercise PIPING SHRIKE ended in the Shoalwater Bay training area recently as realistically as it began.
[…]
Something else to come out of PIPING SHRIKE was the hide of a dreaded Drop Bear, below, nailed to a tree outside the Q Store. The KSLI heard many reports from Diggers about this beast.
ARMY’s full report on the habits and habitat of the Drop Bear appears on p13, this issue.

2-: From Australian Army: The Soldiers’ Newspaper (Canberra, Australian Capital Territory, Australia) of Thursday 9^th December 1976 [page 7, column 4]:

Silver City was missing at the . . .
Silver ceremonial at Kapooka

THE 1^st Recruit Training Battalion, Kapooka, was 25 years old last month.
[…]
With 25 years of history behind the 1^st Recruit Training Battalion, many soldiers who passed out from 1 RTB will remember the legends and stories of drop bears and hoop snakes that supposedly originated there.

3-: From an account of an inter-service shooting competition held at Coonawarra, Darwin, in the Northern Territory, published in the Royal Australian Navy News Pictorial (Sydney, New South Wales, Australia) of Friday 29^th September 1978 [page 14, column 1]:

I-S SHOOTING—THE LIGHTER SIDE:
[…]
Initially a lot of problems were caused by the Drop Bears which are apparently prolific in that particular area, but Sgt Mill finally had them under control and they ceased to actively hinder the team.

4-: From Lily on the Dustbin: Slang of Australian Women and Families (Penguin Books Australia Ltd., 1982), by the Australian author Nancy Keesing (1923-1993) [page 85]:

The ‘drop bears’ are creatures of a tall story—they were invented during World War II for the benefit of gullible American servicemen. It is alleged that ‘drop bears’ are a dangerous kind of koala and that they drop out of trees on the heads and shoulders of bush walkers and hug them to death.

5-: From 21^st Birthdays, in Classifieds, published in The Canberra Times (Canberra, Australian Capital Territory, Australia) of Saturday 31^st July 1982 [page 24, column 3]:

TAM.—Beware of drop bears in the future, for sure, totally love Clint.

6-: From Listening Post (Sydney, New South Wales, Australia) of October 1982 [page unnumbered, column 1]—The reference is to the Drop Bears, a Sydney-based rock band:

Dance!
Three of Sydney’s top bands will gather to help SER subscribers, supporters and workers celebrate a radio rage. The Riptides (pictured), Pel Mel and Drop Bears are the bands and you are the people we want to see at the station’s inaugural broadcasting bash.

[7 to 21-: From various Australian newspapers and magazines, from Monday 7^th March 1983 to Tuesday 7^th May 1985, with reference to the Sydney-based rock band the Drop Bears.]

22-: From Accommodation Vacant, in Classifieds, published in The Canberra Times (Canberra, Australian Capital Territory, Australia) of Saturday 4^th October 1986 [page B14, column 7]:

NEAT person to share Mt Stromlo group hse with kangaroos and drop bears. $32pw plus kitty. Ph 883391.