The hosts will resume on 18-2 on the final day of the fourth Test, needing to bat at least 98 overs to draw.
The five-match series is level at 1-1 after Ben Stokes’ phenomenal century gave England a stunning victory in the third Test at Headingley.
“I’m always positive – we can do it,” said Bayliss.
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“We’ve already spoken about who is going to put their hand up and be the hero.
“It will take a couple of our guys to make good hundreds but, as we saw in the last Test, anything is possible.”
After England narrowly avoided the follow-on to be dismissed for 301, a deficit of 196, Stuart Broad and Jofra Archer reduced Australia to 44-4 in a thrilling afternoon session.
Pat Cummins then removed Rory Burns and Joe Root in consecutive balls before Joe Denly and Jason Roy survived to the close.
“It’s a big challenge but we’ve got two very good players out there and we’ve got some guys in the shed that can make hundreds,” said Bayliss.
“When you lose a couple of wickets so close to the end of play it’s a little disappointing but we are pretty upbeat.
“Hopefully we can bat well, save the game and enjoy a beer after.”
Former England captain Michael Vaughan said to salvage a draw the hosts needed a “miracle” on par with their thrilling win in Leeds.
“To face that Australia attack on this pitch now for 98 overs, it will take an incredible innings,” he told Test Match Special.
“You think it will have to be Ben Stokes again – he is the one player that looks calm and composed.
“There have not been enough of the England team who have arrived – it has been individuals and individual brilliance.”
Australia batsman Smith, who now has 671 runs at an average of 134.2 in this series, said his side will just keep “hitting that top of the stump length” to get the eight wickets they require to retain the Ashes.
Cummins removed Root for a golden duck with a majestic delivery that left the England captain and hit the top of off-stump.
“Cummins hit that length beautifully – that ball to Root is a blueprint of what we need to do,” Smith said.
“It will be difficult for England, particularly while the ball is hard.
“I felt pretty vulnerable in my first 20 or 30 balls when they were bowling that good length and there is enough up and down and sideways movement.”