Of British-English origin, the adjective bonkers means mad, crazy.

EARLIEST KNOWN USE

This adjective is first recorded, in the phrase to go bonkers and apparently as army slang, in Here’s a bloke who likes India, a letter from a soldier who had recently been posted to India, published in the Daily Mirror (London) of Thursday 12^th July 1945; this is the beginning of the letter:

I’ve been in India only two weeks, a fact which makes me self-conscious—particularly when I’m with men who have been here the same number of years and more.
They don’t rub it in though—like the old “troops” at home were fond of doing to us “rookies.”
I suppose they feel different about service out here, and I’m beginning to understand why, although the reason is almost too deep and sentimental to explain.
It wasn’t very long ago that the thought of two or three years’ service in strange Eastern lands, with its even stranger mode of living, gave me the “jitters.”
No more going “up-the-line” every leave; no more “spree” in the local town on pay night; no exciting flirtations—l’m single—with girls.
None of these essentials, which make life for two long years—maybe three.
Strewth! I’d go ‘bonkers’!

And this is the end of the letter:

Yessir! We’re doing plenty of thinking and planning out here, and it’s all to do with happiness which keeps us happy.
Just you keep sending plenty of letters, telling us about everything—any little thing.
Then we can sit down, letters spread out, for a few quiet hours with you.
And if we do that often enough, we won’t lose contact with things… and we won’t go “bonkers”!

The introductory paragraph, presented as an illustrated inset, was:

Most Servicemen writing from abroad have plenty of grumbles about the new life and climate. But here, for a change, is a young soldier who feels differently. He finds there’s a bright side.
You can believe him when he writes he’s thinking of you. That keeps him from going “bonkers”!

ORIGIN

A clue as to the origin of bonkers appears in A dictionary of Forces’ slang, 1939-1945 (Secker & Warburg – London, 1948), by Eric Partridge, Wilfred Granville and Frank Roberts:

Bonkers, light in the head; slightly drunk. (Navy.) Perhaps from bonk, a blow or punch on the bonce or head.

The noun bonk is first attested, in the sense of a blow (on the head), in Cannibal Quest (Hurst & Blackett – London, 1934), by Gordon Sinclair (1900-84), Canadian journalist and author:

One night I was walking with two other men through a cultivated coconut plantation with no more to fear than a casual snake or a bonk on the head from a falling coconut.

The noun bonk might be echoic or related to the earlier noun bonce (itself perhaps related to bounce), first recorded as slang for the head in A Dictionary of slang, jargon and cant (Ballantyne Press – Edinburgh, 1889), by Albert Barrère and Charles Godfrey Leland:

Bonce, the head, called also “crust, chump.” From bonce, a marble of larger size than ordinary, used by boys. The French slang for head, bille, literally a marble, bears out this derivation.

If bonkers is indeed from bonk, then it is comparable to the following adjectives formed with the suffix -ers:
– crackers, attested in 1928, meaning insane, probably from cracked in the sense insane;
– ravers, attested in 1938, meaning, and from, raving (mad);
– starkers, attested in 1956 in the sense completely mad, from stark (mad).