New Zealand skipper Kane Williamson has come under fire as he and his team-mates wilt in the Perth heat.

Williamson didn't 'walk' after getting a faint edge to a delivery from Australian paceman Pat Cummins in the final session of the first test.

Only former Australian captain Steve Smith was keen to discover if Williamson had edged the pink ball through to wicketkeeper Tim Paine while on 13 as he and Ross Taylor battled to get New Zealand back on track after a horror start to their first innings.

AP Australia's Steve Smith, third left, celebrates with teammates the dismissal of New Zealand's Kane Williamson in the first test in Perth.

​Snicko showed a tiny spike when ball passed bat, enough to give Williamson out caught behind, and his poker face was priceless on replay. Smith later got the perfect revenge when he grabbed a sensational catch in the slips cordon to dismiss Williamson after he'd added another 21 runs.

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But Williamson was slammed by some for not giving himself out, with a few of his critics drawing attention to the apparent hypocrisy of the Black Caps recently winning the ICC's Spirt of Cricket award.

As we scramble around desperately for any tiny crumb of good news, praise be to Spirit of Cricket hero Kane Williamson for not walking again. https://t.co/dNlW6PiWRg — Dave Tickner (@tickerscricket) December 13, 2019

New Zealand were given the award for their sporting conduct during and after the tied World Cup final, which was controversially won by England on a boundary countback tiebreaker at Lord's in July.

"The New Zealand team are worthy winners of this award," MCC president Kumar Sangakkara said.

GETTY IMAGES Kane Williamson came under criticism for not walking.

"In the heat of battle, they displayed a level of sportsmanship that was fitting for such a fantastic final, and indeed tournament.

Former winners of the award - created in 2013 - include ex-Black Caps captain Brendon McCullum in 2015.

Kane Williamson not walking off AGAIN. So much for the spirit of the game. Smh — N (@yourstruly49_) December 13, 2019

Williamson got some sympathy over being the victim of Smith's sensational snare by the victorious bowler, Mitchell Starc, who ripped through NZ's batting to leave them stumbling at 109-5 in reply to Australia's first innings of 416.

"Unfortunately, for him, you sort of expect that from Steve. He has done that a few times," Starc said.

"That is a beauty, one of the best he'll ever take given the circumstances, the player, the light, pace of the ball," former Australian batsman Mark Waugh said on the television commentary.