The Australian-English phrase to spit the dummy, and its variants, mean:
– to lose one’s temper or composure;
– to give up contesting or participating.

The image is of a baby spitting out its dummy in a tantrum.

These are, in chronological order, the earliest occurrences of the phrase to spit the dummy and variants that I have found:

1-: From Unpaid fines can mean arrest, by Inspector Ron Dillon, published in The Canberra Times (Canberra, Australian Capital Territory) of Thursday 2^nd August 1979—the author plays on the literal and figurative meanings of the phrase, which is in quotation marks:

The arrest sheets of the last few days show 11 individuals taken into custody for non-payment of fines. This is not an unusual number as arrests for failing to pay fines are virtually a daily occurrence, but I thought this item might enlighten the uninitiated and save some embarrassment of being escorted off the lockup to wait while the “better half” raises the cash.
[…]
Oops. Noticed on our arrest sheet that a young man born on 28/7/1979 was charged with breaking, entering and stealing. He probably “spat the dummy out” when taken into custody.

2-: From the column Case history, by Bill Casey, published in The Sun-Herald (Sydney, New South Wales) of Sunday 11^th January 1981:

Gary Gilmour spat out his dummy this week, complaining he hadn’t been given a go by the NSW cricket selectors.
“I’m heading interstate next season,” said big Gus. “It’s obvious I’m not going to get another chance here.”

3-: From the column Casey looks at life, published in The Sun-Herald (Sydney, New South Wales) of Sunday 8^th February 1981:

Find your rating—at your peril

I am going to form a society for the banning of newspaper and magazine “Figure Your Rating” quizzes.
I know the Great Smoking Demon Thunderclap called Editor will spit his dummy fair in the corner of his teepee when he reads this, but it cannot be helped.

4-: From Brisbane pull out of Cup, by Alan Clarkson, published in The Sydney Morning Herald (Sydney, New South Wales) of Tuesday 19^th July 1983:

Mr Ron McAuliffe, the president of the Queensland Rugby League, last night withdrew the Brisbane team from the midweek KB Cup competition and threatened that Queensland would quit the Australian Rugby League.
[…]
“They must think we are bonkers,” an irate Mr McAuliffe replied after being told that the NSW Rugby League had voted unanimously to stage a proposed KB Cup semi-final match between Brisbane and Manly at Leichhardt Oval.
“[…]
“We were told we were to have three KB Cup games at Lang Park during the season and we were given the choice of having either the quarter-final or the semi-final here.
“The Brisbane League decided they would have the semi-final at Lang Park in accordance with the agreement with the promoters and in discussions with the former president, Kevin Humphreys.
“Now, Manly has apparently spat out the dummy, called in the reserves and pushed through a move that was against the original agreement.”

5-: From an article by Peter Bowers about the visit to the Middle East made by Bill Hayden (born 1933), Minister for Foreign Affairs and Trade, published in The Sydney Morning Herald (Sydney, New South Wales) of Saturday 11^th February 1984:

When Labor came to power last year, the Israelis could be forgiven for thinking that, given Mr Hawke’s ¹ strong personal commitment to Israel, his Government would be more sympathetic towards them than the Fraser Government ².
However, an early review of Middle East policy produced a firmer position on the right of self-determination and independence for the Palestinians and called for a freeze of Israeli settlement in the West Bank. […]
[…]
Had the Fraser Government remained in power it might have made similar adjustments but as Labor made the changes the Israelis see the Hawke Government as less sympathetic towards it.
For Israel, a country running out of friends, marginal differences and subtle nuances are important. Mr Hayden’s four-day visit did nothing to allay Israeli uneasiness. Nothing was said to Mr Hayden (“I didn’t see anyone spit the dummy”) but reporters travelling with him got bad vibes from the Israeli public relations level.

¹ Bob Hawke (1929-2019) was the Prime Minister of Australia from 1983 to 1991, as Leader of the Australian Labor Party.
² Malcolm Fraser (1930-2015) was the Prime Minister of Australia from 1975 to 1983, as Leader of the Liberal Party of Australia.

6-: From ‘A bit of a bore’ surprises at Sandown: Colt makes amends, published in The Canberra Times (Canberra, Australian Capital Territory) of Sunday 29^th April 1984:

Melbourne: Colac trainer John Sadler admitted that his colt Bute Street was a bit of a bore until yesterday.
Bute Street had had so many setbacks in his preparation that Sadler had resigned himself to yet another disappointment.
But Sadler was prepared to forgive and forget after the colt won the Macalister Handicap, of 1,400m, at Sandown yesterday. […]
[…]
To find a reason for Bute Street’s improvement, Sadler said the colt had not been put under extreme pressure yesterday.
“That may have made the difference,” he said. “He has had his chance in several races but seems to spit the dummy out when the pressure is on.”

7-: From The light at the end of the Budget, by Geoffrey Barker, published in The Age (Melbourne, Victoria) of Saturday 25^th August 1984:

Mr Keating ³, who went to inordinate lengths to attack Opposition economic policies, lashed out hard and personally at Mr Peacock ⁴. “A couple of months after taking over the industrial relations portfolio, he spat out the dummy because it was too hard for him. When he held the foreign affairs portfolio, he was always swanning around. In his whole parliamentary career, he has never touched a hard domestic issue. . .”

³ Paul Keating (born 1944) was the Treasurer of Australia in the Hawke Government from 1983 to 1991—cf. note ².
⁴ Andrew Peacock (1939-2021) was then the Leader of the Opposition, as Leader of the Liberal Party of Australia.