The Australian-English phrase flat out like a lizard drinking, and its variants, are humorous extended forms of flat out, meaning with the maximum speed or effort.

In this phrase, there seems to be wordplay on flat out, meaning lying stretched out.

These are, in chronological order, the earliest occurrences of the phrase flat out like a lizard drinking and variants that I have found:

1-: From the account of a cricket match between Uralla and Public Service in the New England district competition, published in The Uralla Times (Uralla, New South Wales) of Monday 10^th February 1930:

The home team’s prospects began to brighten. For anything that offered runs they went flat out, like the lizard drinking, and knocked them up in fine style.

2, 3 & 4-: From The Land (Sydney, New South Wales):

2-: Of Friday 23^rd May 1930:

Our Simile Competition

Slowly but surely similes are attracting attention, and before the competition closes our language should be the richer by many quaint and humorous expressions. Here are this week’s contributions:—
A grin like a goat eating a cabbage.—Pyramus.
Flat out, like a lizard drinking.—Daring Demon.
Toddling like a hobbled grub.—Daring Demon.
Green as a new chum—Spare Part.
As ragged as Oxford Bags.
As happy as the Hive.
As lovesick as Ned Kelly.
As daring as Dauntless.
(The last four by Painkiller.)

3-: Of Friday 30^th May 1930:

More Similes

[…]
Going “flat out” like a Bee after a new subscriber.
[…]
Flat-out, like a jew [sic] lizard on a rail.

4-: Of Friday 6^th June 1930—from Country Mirror: News Items from “Land” Readers:
—as deciphered from the original document, which is almost illegible:

Wirrinya […]: This last week has been ideal for cropping, and most of the farmers have been going flat out like lizards drinking; a few have finished. We have lights on the tractor and have been working day and night, three shifts—that’s the way to cover the ground.

5-: From the account of a rugby match between Kellerberrin and Tammin, published in The Eastern Recorder (Kellerberrin, Western Australia) of Friday 11^th July 1930—the use of the adjective proverbial indicates that the phrase was already well established:

At the bounce the home team, spouting water in all directions swam for the visitors’ goal. At a critical moment Gallagher threw the ball and Captain Steele’s passionate outcry qualified him for a seat on the Trades Hall Committee. But Tammin were flat out like the proverbial lizard drinking, and managed two singles in rapid order.

The phrase flat out like a lizard drinking has occasionally been used in the opposite sense—as in the following from Truth (Sydney, New South Wales) of Sunday 15^th November 1931, in which flat out means totally exhausted:

BUTCHERED
Frank Woodham’s Cash
KING MOAB LICKED

Big boy butcher Woodham will have to sell quite a few juicy undercuts during the week to get back what he lost on his colt King Moab in yesterday’s Woollahra Stakes.
The Moabite colt was regarded by the stable as a put in and take out affair; even when he drew No. 14 post position they didn’t worry.
“Could win from anywhere” was their summing up, and in went the big chops and steaks on His Royal Highness.
When he beat Tingalba in the two-year-old classic, King Moab came from a long way back, and was very wide out.
So when he jumped away clearly yesterday, and was sitting pretty in fourth place at the end of the initial furlong, Woodham thought the fillets were on the fire.
Darby Munro had a choke hold on the colt as he swung for home, and it certainly did appear that King Moab would go like a scalded cat the instant Munro took the pressure off.
Darby gave him his head two furlongs from the post, but King Moab, instead of dashing right away, commenced to totter and lose ground.
There was nothing like his Breeders’ Plate finish about him yesterday, and when he passed the post in fifth place, King Moab was flat out like a lizard having a spot.