The colloquial phrase (as) drunk as a pissant means: extremely drunk.
—Cf. also the phrase (as) game as a pissant, meaning: plucky, courageous.

The Australian author Stephen Murray-Smith (1922-1988) made a hypothesis as to the origin of the phrase (as) drunk as a pissant in the following from Right Words: A Guide to English Usage in Australia (Ringwood (Victoria): Viking, 1989)—as quoted in the column Passionate Reader, published in The Age (Melbourne, Victoria, Australia) of Saturday 29^th April 1989 [Saturday Extra; page 6, column 1]:

The meaning of pissant. Our use and understanding of the word is that there is a species of (Australian?) ant that is small, aggressive and, even more than most ants, smells like urine when crushed. Thus a human who is called a pissant is regarded as small and unimportant but noisy and aggressive (and possibly able to leave a smell behind when crushed?).
[…]
To pissant around follows clearly enough. So does brave as a pissant. But drunk as a pissant is not so clear. Perhaps the aggressive behavior of pissants is reminiscent of drunks waving their arms about on the streets.

The earliest occurrences of the phrase (as) drunk as a pissant and variants that I have found are as follows, in chronological order:

1-: From Burrowings, by ‘Wombat’, published in The Woodend Star and Newhamshire, Macedon, and Gisborne Advocate (Woodend, Victoria, Australia) of Saturday 5^th November 1892 [page 3, column 4]—the phrase is as drunk as an ant, and, although ‘Wombat’ wrote that it was “a very common expression”, this is the only early occurrence that I have found:

Talking about “spirits” brings to my mind a sight I saw in Woodend on the evening of the day on which the Woodend Hunt Club held their point-to-point steeplechase. At a spot in the town—I wo’nt [sic] particularize it—I saw a boy who was, to use a very common expression, “as drunk as an ant.” Why this expression is applied to the ant is beyond my comprehension, as from observation I have long ago arrived at the conclusion that the ant is an industrious little creature. Many a time I have sprinkled sugar across a track made by ants, and watched them carry it to their homes. Perhaps it is because people liken the industry of the ant to the constancy of the individual who raises his elbow with such celerity, in order that he may not be found backward if he is asked to “have one?” by a friend. However, the comparison is made, and I think it a very unjust one—to the ant.

2 & 3-: From two novels by the U.S. author John Dos Passos (1896-1970):

2-: From Manhattan Transfer (New York: Harper & Brothers Publishers, 1925) [Second Section, chapter 7, page 250]:

“Hey quit dat.” “Fellers lets trow dis guy out. . . . He aint one o de boys. . . . Dunno how he got in here. He’s drunk as a pissant.” Stan jumps with his eyes closed into a thicket of fists. He’s slammed in the eye, in the jaw, shoots like out of a gun out into the drizzling cool silent street.

3-: From The 42nd Parallel (New York: Harper & Brothers Publishers, 1930)—as reprinted in U.S.A.: 1. The 42nd Parallel; 2. Nineteen Nineteen; 3. The Big Money (Boston: Houghton Mifflin Company, 1937) [page 76]:

When he woke up Ike was sitting on the bed. Ike’s eyes were bright and his cheeks were red. He was still a little drunk. “Say, Mac, did they roll yer? I can’t find my pocketbook an’ I tried to go back but I couldn’t find the apartment. God, I’d have beat up the goddam floosies . . . Shit, I’m drunk as a pissant still.”

4-: From The Australian Language (Sydney and London: Angus and Robertson Ltd., 1945), by Sidney John Baker (1912-1976) [chapter 4: The Bush; page 87]:

Here are a few more similes snatched from our environment: […] game as a piss ant or drunk as a piss ant (we have taken a verb from this term: someone is pissanting around when he is messing about, and we pissant someone when we defeat or outwit him).

5-: From Riders in the Chariot (London: Eyre & Spottiswoode, 1961), by the Australian author Patrick White (1912-1990) [chapter 13, page 448]:

“And on such a day!” she shrieked, looking at the clock. “I bet that nephew of yours will be full as a piss-ant by eleven!”

6-: From Nicely, thank you (Drunk 2000 times): A Frolic with some Synonyms (Melbourne: National Press, 1971), by the Australian polymath and bon vivant Oscar Mendelsohn (1896-1978) [page 28]:

Drunk as an ant
Drunk as an owl
Drunk as a pig
Drunk as a piper
Drunk as a pissant
Drunk as blazes
Drunk as Chloe

7 & 8-: From Stribling (London: W. H. Allen, 1974), by the U.S. author Stephen Longstreet (1907-2002):

7-: [chapter 17, page 110]:

“What’s the matter?”
“The matter,” she said, “I’m drunk as a pissant, that’s what’s the matter, and these little agent bastards want me to do some standards on the soup hour here I wouldn’t look at sober.”

8-: [chapter 35, page 257]:

“Old Luke he got drunk as a pissant at the Trout Stream Bar and Grill. Loaded.”