Does Australia have heatwaves?

Yes. Australia is prone to heatwaves during the summer and occasionally even late spring and into early Autumn as well. While the heatwaves are most likely to hit the southern states and the west, they have been known to hit northern centres.
The town of Marble Bar in Western Australia holds the world heat record, having experienced a prolonged heatwave in the early 20th century. On that occasion, Marble Bar recorded maximum temperatures equalling or exceeding 37.8 degrees Celsius on 161 consecutive days, between 30 October 1923 and 7 April 1924.
In 1896, a heatwave through New South Wales resulted in almost 200 deaths.