During the Three Kingdoms? The only real possible candidate is Song Jian that I can think of.

Song Jian was probably part of the Liang rebellion that, between the Turbans and the civil war starting (but which the novel ignores for several potential reasons), the Han lost control of Liang. Song Jian, who may have been non-Chinese or was a local man, based himself in his home area of Fuhan, at a tributary of the Yellow River (north of present-day Lanzhou) and proclaimed himself King of the Sources of the River Who Will Pacify Han, taking up imperial rituals and appointing his own officials. With Liang divided among various local leaders and warlords, he managed to maintain his independent state for a considerable period.

It ended in the winter of 214, by this point the Liang leaders had been destroyed by or surrendered to the controller of the Han Cao Cao, leaving no protection other than distance for Song Jian's regime. Cao Cao's kinsman/general Xiahou Yuan led the siege for a month and he took Fuhan with Song Jian executed (or burning himself to death) and his officials executed, Yuan sent Zhang He to Heguan and other areas to pacify. Cao Cao however did not restore an administration at Fuhan, Song Jian was outside the limits of the capacity of the warlord state.

Not much is known about Song Jian and his regime, the records are small (usually about Liang revolt then his fall) and can be conflicting. Possibly of the Yuezhi, he was in a distant area, cut off from the central plains and any sense of Han authority, shielded by a wall of other local leaders from any figure who might seek to punish his revolt for decades. The novel Romance of the Three Kingdoms leaves him out.

There was a plan to split China in two, in theory. In 229 Sun Quan became the founding Emperor of Wu and allies Shu-Han sent minister Chen Zhen as an envoy to congratulate. Their seething at Sun Quan saying Han had lost the mandate was to be chocked back. There, perhaps on Sun Quan's initiative, a blood oath composed by Hu Zong would be sworn (and done likewise by Shu-Han's chief minister Zhuge Liang) that divided the land. Each got to keep their own lands and from the larger Wei lands under Emperor Cao Rui, Shu-Han had a claim to the northern lands of Bing, Liang, Ji (and also Yan of the Gongsun family, something Wu perhaps not so keen to bring up when later seeking an alliance with Gongsun Yuan). Wu got the southern and central lands of Xu, You, Yu and Qing while the old capital region of Sili was split with both sides getting one old Han capital each (Chang'an for Shu-Han and Luoyang for Wu).

It was a message to Wei and the populace across China that the alliance was firm, that they recognized each other's legitimacy including that of Zhuge Liang as chief minister and were going to destroy Wei. There were no hard feelings of the seizure of Jing in 219 or the events that followed including Sun Quan's argument Han had exhausted the mandate, they are blood allies. It also meant Shu-Han pledging not to interfere in that area again but focusing their efforts via Hanzhong through the north and north-west. However, the arguments still appealed to a shared history, of an empire.

A few years earlier another Shu-Han envoy Deng Zhi had gone to Sun Quan. The then King suggested it would be happy days when there would be just the two powers left and there could be peace but Deng Zhi bluntly argued there could not be two suns in Heaven nor two such figures on earth and there would then be war if Sun Quan did accept Shu-Han as the true rulers. That was likely the truer feelings of both sides than the diplomatic and public sharing of 229 and it was not seeking to found new countries but to share the lands of China.

I know of no other example. Be it in the northwest with the Gongsun family of Liaodong, to the south with Shi Xie and family in Jiao, the west in Yi or the warlords in the central plain. This may partly be that those outside the warlords, the rebels and bandits, were poorly recorded and if one was going "Xu independence", that got neglected in the records.

There was certainly regional identity, even in the way the records of the three kingdoms writer Chen Shou promotes native scholarship of Yi and has a go at some of the Jing scholars in the Shu-Han court. It impacted policies as Liu Biao swung away from his northern Jing advisers to those from the south of his lands, or the destruction of the power in northern families of Wu with the falls of Zhuge Ke and Teng Yin or the consideration (rejected) on Cao Cao's death on installing only those from Cao family areas of Qiao and Pei as head of cities. Areas had their own customs, heroes and their own tales, regional history was starting to become a thing with the likes of Qiao Zhou exploring the landscapes and tales of his Yi home and others like Yu Fan promoting local heroes.

In some areas, the local gentry probably did benefit from no central control by a distant court and welcomed the chance for using resources locally, happy to push their warlord to not submit. While not perhaps also being as keen to push their resources into fighting to unify the land but that isn't the same as their thinking "we are a seperate country" or that the land under heaven wouldn't be unified someday, the clash between those pushing to expand imperial power via war and those wishing for the court to concentrate on local and cheaper affairs was far from unknown during the Han.

China had gone through a long united period under the Han (and a Wang Mang interruption), during which some of the more energetic administrators would try to impose some Han culture on the locals to strengthen Han grip. They had lived under shared systems for generations while, though were local heroes, there was a shared history with shared figures, texts and shared concepts like the mandate of Heaven.

That didn't go with the collapse of the Han's authority. For a warlord to break away from that and why nobody (bar Song Jian) did so hasn't been studied to my knowledge but I can suggest a few potential reasons why a warlord might remain within the system, apart from their own cultural ties. Warlords themselves were not always from the local area having been appointed from the outside (though in itself not a deal-breaker). Within their courts would be local gentry families but the locals themselves might not be united and the refugee scholars there, sometimes providing important support to the warlord, who might not be too pleased to find themselves in a foreign land. Would this breakaway lead to the other regional powers turning on the leader as they did Yuan Shu when he claimed to be Emperor in 197 CE? A ruler marking himself out as a traitor may also be limiting his future options for negotiation if things turn bad.

I hope this helped

Sources:

Records of the Three Kingdoms by Chen Shou

Comprehensive Mirror for Aid in Government by Sima Guang, translations by Achilles Fang and Rafe De Crespigny

The Talent of Shu: Qiao Zhou and the Intellectual World of Early Medieval Sichuan by J. Michael Farmer

Northern Frontier: The Policies and Strategy of the Later Han Empire by Rafe De Crespigny

The Life And Legacy Of Liu Biao: Governor, Warlord and Imperial Pretender by Andrew Chittick

Generals of the South: The Foundation and Early History of the Three Kingdoms State of Wu by Rafe De Crespigny