I've picked three things I associate with the Wild West: weak government, Theft of native land, and gold rushes. New Zealand had all three in the nineteenth century.
Weak government: Before 1840, there was no central authority in New Zealand. Instead the country was divided between competing iwi, and below that into hapu. Think of iwi as tribal confederations, and hapu as individual tribes. In that period, enterprising Pakeha (ethinic Europeans) traded muskets and metal tools to iwi for timber, flax, alcohol, prostitution services and foodstuffs. I mention prostitution and alcohol because New Zealand's first international port, Kororāreka, was known as the "hellhole of the pacific" because of its reputation as a whaler's paradise of boozing and whoring https://russellnz.co.nz/old-history/a-hell-hole/. The arrival of muskets and the proliferation of new crops expanded the ability of iwi to engage in warfare, leading to the musket wars. The musket wars were a serious of invasions, spreading south across New Zealand as new tribes acquired muskets, and used them on their weaker neighbours to the south. They began when Ngāpuhi warlord Hongi Hika led a campaign of revenge against his Ngāti Whātua neighbours. Hika was an interesting figure, among the first adopters of the musket in New Zealand, and the toast of London's high society during his 1820 visit. https://teara.govt.nz/en/biographies/1h32/hongi-hika
https://teara.govt.nz/en/musket-wars gives a good summary of the musket wars. In essence, the ritualised seasonal warfare of pre-contact Maori gave way to total war and bloody conquest. European traders and sailors often interfered by towing Maori war Waka (canoes also known as Taua), or by selling muskets and metal weapons to iwi - much to the consternation of the crown.
Theft of native land: In 1840, representatives of the British crown signed a treaty with most tribal leaders in new Zealand, first in Waitangi and then touring the country. The treaty is too much to explain here, and is politically contentious to this day. I'll simply say that it was probably organised with good intentions to minimise settler Maori disputes, and bring criminal Brits in New Zealand under the Queens heel. However, it opened the door for mass European migration which led to further instability and the mass theft of native land. Most famously, the two versions of the treaty (one maori one english) were not perfect translations. In the English version, Maori were said to give "sovereignty" over the land to the crown, whereas in the Maori version the term was "Kawanatanga" which is closer to governership. Regardless of which version you read, it was widely ignored in the period from 1860 (when Pakeha population passed Maori) to around 1940, as Maori were alienated from their land in a series of seizures and shady purchases. Increased settler control of the government probably made this worse - the distant British crown was a relatively even handed arbiter compared to voracious settler governments. Maori did not take these land seizures lying down, and a series of conflicts now known as the New Zealand land wars shook the country between 1840 and around 1870. Many Maori fought on the crown side in these wars, but this loyalty was rarely rewarded.
Gold rushes: Our main gold rush was down south. Around 18,000 prospectors came to New Zealand between 1861 and 64 to prospect for gold in Otago, and many of them never left. Among them were the first Chinese immigrants to New Zealand, the ancestors of New Zealands 'old Chinese' community.
I've missed a huge amount of stuff, including: Railroads, religious movements and awakenings, a maori king movement that persists to this day, details on the New Zealand land wars and the musket wars, Hongi Hika's story, the chopping of the flagpole, details of the native land court and the seizure of the Waikato, the invasion of Chathams, a New Zealand trojan horse (the elizabeth affair), governor George Grey, the missionaries, non-violent resistance at Parihaka and lots more.
If you want to know more, take a look around Te Ara, the Encylopedia of New Zealand https://teara.govt.nz/enor https://nzhistory.govt.nz/and here's a link to James Belich's documentary "the New Zealand wars" it's a little dated, but he was good scholar and a passionate historian: https://www.youtube.com/watch?v=WFOkgIAjXiwand just search up our nineteenth century history, it was all very wild west.
Edit, a word or two.
To add to the theme of gold rushes, brought up in /r/Bokpokalypse’s post. There was quite a lot of cross-contamination from North America in terms of goldmining technologies and the people that brought them by various routes across the Pacific. The most common would be for a man perhaps just making enough to scrape by on the Californian goldfields post-1848 to hear of rich rewards to be had in the newly-discovered Australian goldfields. A few years later, upon hearing of even richer rewards to be had in inland Otago, New Zealand our brave American miner, if he had earned enough to afford it, would have moved himself as fast as possible across the Tasman sea to join the rush to places like Gabriel’s Gully or the Arrow River. These men brought with them the accumulated knowledge and experience of 10 years or more on the American and Australian goldfields to the newly discovered goldmining areas of New Zealand.
It’s important to not understate the scale of the rushes that occurred in terms of the movement of men and materials. Once news of a discovery was out a veritable hoard of men would swoop down into an area. This touches on another of the themes mentioned in the other post – lawlessness. When whole towns-worth of people, comprising nearly exclusively of men, appeared overnight, the pressure of space – to pitch ones claim for prospecting – and the lack of infrastructure or support to actually feed this multitude of people often over-boiled into violence that was not checked by any police force; as none had been yet assigned to that area. Often the authorities would have to beg other colonies for aid in policing. As was common in Australia, often it was down to the miners to police themselves, with local committees enforcing a code that could see offenders summarily punished or expelled. This code often ‘included a prohibition on strong liquor, an informal but decisive disputes tribunal, and methods to ensure essential supplies and stores were properly and fairly distributed.’ (Singe, Miles & Thomson, 1992).
Even when police had become established, their main task was to provide an escort for the gold and scant resources were available for anything more. At Arrowtown between 1863-65 apprehended miscreants were often just chained to a large log in lieu of a lockup building being available - leading to one memorable moment when one such prisoner ‘a gigantic Irishman’ strode back into town with the log under his arm and was found drinking heartily in a hotel!
Main sources for the above are the works of John Hall-Jones, principally his Goldfields of Otago. More on Policing in the early period was from Singe, Miles & Thomson’s Authority to Protect - A Story of Policing in Otago.