The colloquial New-Zealand and Australian expression flat stick means: at full speed.

In this expression:
– the first element is perhaps after the expression flat out, meaning: with the maximum speed or effort;
– the second element is perhaps after the adverb quicksticks, meaning: quickly, without delay.

—Cf. also the Australian-English phrase flat out like a lizard drinking, meaning: with the maximum speed or effort.

The expression flat stick originated in New Zealand—the earliest occurrences that I have found are as follows, in chronological order:

1-: From The Press (Christchurch, Canterbury, New Zealand) of Tuesday 10^th February 1970 [page 1, column 1]:

Sailing To Holland

A young Christchurch man is building a 42ft concrete yacht in a vacant section in Merivale. He plans to launch it in April and to sail it to the Netherlands, his homeland.
Mr L. Ooms has been building his yacht for the last two years. He has been in New Zealand for the last six years. “I am going flat stick to get it finished now,” said Mr Ooms.

2-: From The Press (Christchurch, Canterbury, New Zealand) of Saturday 11^th December 1971 [page 1, column 3]:

BOLTING HORSE STOPPED

A young man jumped from a fast-moving truck yesterday on to the back of a bolting horse which had earlier thrown its rider in Wainoni.
Mr Trevor Waihi, aged 17, of 53 Wainoni Road, and a companion, Douglas Martin, aged 15, noticed the horse running towards them in Wainoni Road.
[…]
“The horse had panicked and was going flat stick down the middle of the road,” Mr Waihi said.

3-: From The Press (Christchurch, Canterbury, New Zealand) of Tuesday 26^th December 1972 [page 2, column 6]:

Quiet Christmas
(N.Z. Press Association)

AUCKLAND. December 25.
For ambulance officers and the police, Christmas Eve on a Sunday was the best Christmas present they could have received.
“It was the quietest Christmas Eve I can remember,” said one ambulance officer. “Quite obviously it was that way because the pubs were closed.”
[…]
A senior police officer supported these comments.
“It’s been exceptionally quiet in the way of accidents,” he said. “We had no trouble at all on Sunday. But you can bet your bottom dollars that if pubs had been open we would have been flat stick giving breath tests.”

4-: From The Press (Christchurch, Canterbury, New Zealand) of Thursday 4^th October 1973 [page 1, column 10]:

Weightlessness at wheel top
(N.Z. Press Association)

PALMERSTON NORTH, October 3.
Under certain conditions a ferris-wheel cage could become weightless and tip over, an inquiry in Palmerston North was told today.
[…]
The inquiry is being held before a retired New Plymouth Magistrate, Mr A. W. Yortt, and a Lower Hutt consulting engineer, Mr A. G. Gilbert, to find the cause of death of two girls and the serious injury of another.
[…]
Evidence that the wheel was being driven fast was given by Christopher Samuel Morgan of Palmerston North.
He told the inquiry he was near the wheel for about half an hour before the accident.
“The operator would have it going flat stick, then he would jam the brake on, then away it would go again. It was like that all the time.”