Yes, very much so, although for us it isn't called 'the wild west'. There isn't really a name for the period from the 1850s to 1900 in Australia, but the closest might be the Gold Rush or Federation era.

In the 1820s and 30s, Australia was very much an uncolonised land, with the exception of the area around Sydney and the island of Tasmania. Sydney was founded in 1788 as a penal colony, and Tasmania (then known as Van Diemen's Land) was for reoffenders/max security prisoners. As ex-convicts and free settlers spread out from Sydney and throughout Tasmania, to exploit enormous green pasture lands created by Indigenous fire-stick farming, they warred with Aboriginal people, mostly wiping them out. In Tasmania, this led to 'the Black War' and the 'Black Line', which was an attempt at walking across the entire (massive) island in a straight line of men, shooting any Aboriginals they met. Despite only catching an old man and two boys, the Aboriginal people of Tasmania agreed to a ceasefire and relocation to the small and desolate Flinders Island, where they were mostly wiped out by disease.

At the same time, the city of Perth and the Swan River Colony were founded in Western Australia - it stagnated due to a lack of fertile land and severe isolation. Attempts were made at founding settlements in the Northern Territory, but disease and fierce Indigenous resistance meant they ended in failure. Brisbane and Melbourne were also founded in this era - Brisbane as a max security penal colony (which didn't last long until settlers demanded entry) and Melbourne as an illegal sheep and whaling settlement later absorbed into NSW.

That is what you might call 'the British era' or 'the Convict era'. Redcoat soldiers and convicts, the founding of muddy little outposts on a very blank map, and wool as the resource men kill each other for.

This all changes with the discovery of gold in the 1850s. People suddenly come flooding into the country from every part of the British Empire and the USA. Like the Californian gold rushes, Chinese migrants also came in droves, and just like in the US, they were treated quite poorly.Much like San Francisco or the railroad towns of the US, the colony of Victoria suddenly burst to life with colonial activity - Melbourne almost instantly became one of the largest and richest cities in the world, with one of the highest standards of living, and splendid architecture to prove it. Mining towns like Bendigo and Ballarat were much like the Deadwoods of the US, in that it saw frequent lawlessness.

Miners, some of whom were the first generation of white Australians, rebelled against overbearing police and government, leading to the Eureka stockade rebellion, which was a 'battle' between miners and police - the Eureka flag is an important cultural icon of Australia. Pistols and revolvers were incredibly common on the goldfields (many migrants brought US-made Colts from London), and gunshots were frequent on the streets of Melbourne, according to the newspapers of the time.

Much like the US, outlaws like Jesse James were also common. These were called bushrangers, and they preyed on those travelling the vast empty wilderness between towns and cities, often robbing gold wagons, post wagons or stage coaches. They became famous and celebrated, and stole horses and cattle, and robbed banks. People wrote songs and poems about them, and cheered them on, treating them like Robin Hood figures.

The most famous is Ned Kelly and his gang - Americans might known him for his gang's use of armour, but in Australia he is also a potent symbol of anti-authoritarianism, a freedom fighter. For a country that barely engages with its history, its incredible how many books and films we've about him - the world's first feature film was a Ned Kelly movie made in Australia in 1906, and there was a decent Ned Kelly film made with Heath Ledger in the mid 2000s, in case that interests you.

When alluvial mining ended in eastern Australia, it started up over in the west, with new discoveries made in remote areas in the north and south of Western Australia. Those with enough money left the east for the west, and Kalgoorlie and Coolgardie sprang to life in the deserts of Western Australia, briefly becoming some of Australia's largest and wealthiest towns. Perth went from a stagnating backwater into a thriving port city, with newly wealthy businessmen tearing down all of the drab wooden buildings of the first 60 years of settlement, and replacing them with grand hotels and theatres. A process Perth has suffered repeatedly with its mining-heavy boom-bust economy.

Easterners from Victoria and NSW brought their colonial cultures and politics with them, uniting the country in a way that geography mostly inhibited (WA was/is a fairly conservative state, and the common man had no vote until eastern miners migrated over and threatened mining profits with political and industrial action). Experts from across the British empire came to build a modern harbour, dams and massive desert pipelines for water. Visit any of Western Australia's Goldfield towns today and you will see very similar architecture to American frontier towns.

Like the US, trains and telegraph lines were being built throughout the states to link the coasts through the sparsely populated Indigenous centre. The explorers and surveyors who had to find the routes for this infrastructure starved in the desert and were speared in Indigenous attacks, and were celebrated for their skill and endurance. Not long after the interior explorers came the 'ranchers' (in Australia called squatters or pasturalists), who usually enslaved Indigenous people (similar to African American slavery) and destroyed the landscape with European farming practices.

Just like in a Western film, you'd have rich landowners dominating the local police and forming posses to massacre natives and hunt outlaws, a law-unto-themselves. Like the US, natives sometimes managed to fight back - Jandamarra was an Aboriginal man who led a guerilla war against white landowners in the Kimberley, still a very wild place today. Aboriginal people were pushed onto reservations, their rights stripped away, their culture outlawed and segregation strictly enforced. The north of Australia is still very 'cowboy' in a lot of places, relying on the cattle industry.

Australia and the US also have similar east-west splits - the east is older, wealthier, greener, more populated, more powerful and more civilised, whereas the west is wild, savage, desert, and lawless. The difference is that Australia also has this in a 'wild' north-'civilised' south split as well, with the middle being mostly 'empty' desert.

The end of Australia's 'wild west' era came with Federation and industrialisation. Federation was the union of the six colonies into the Commonwealth in 1901, which shifted Australia's cultural mindset away from purely local/state matters to thinking of itself as a nation (of Britons) and a part of Empire. Lawlessness died down and agriculture again outshone gold mining in the Australian economy. States became less adversarial, and common cultural practices spread further. Every corner of the map was coloured and linked to a city, even if Indigenous Australians did still outnumber whites in the remote north.

The purity of nation was enforced with legislated attempts at assimilationary genocide like the 1905 Act, which led to the Stolen Generations, and the White Australia policy, which 'banned' non-white migration to Australia. This was also an important time for industrial relations and democracy - Australia was one of the most progressive countries in the world for worker and citizens rights, with voting for women, universal franchise, and strong unions backed by the world's first Labor party.

Australia's 'wild west' era would soon be almost entirely forgotten in the wake of the First World War, which has become the most important time period in Australian popular culture, due to the legacy of the ANZACs. This is partly due to the trauma of the war and the national pride in our involvement on the world stage, but it is also due to modern conservatives pushing white nationalist and jingoist narratives, and pumping huge amounts of funding into ANZAC commemoration (partly to help Australians forget Indigenous history and activism).

Many modern Australians downplay our own culture (a concept called the cultural cringe) and hate it when Aussies act like Americans, so most would see celebration of 'the Australian Wild West' as a terribly shameless attempt at Americanising Australian culture.

Good sources:

The Rise and Fall of Ancient Australia, by Geoffrey Blainey

A History of Australia, by Peel and Twomey.