Yes, absolutely. Since I assume you're referring to the 'age of imperialism', the first and foremost example would be Japan.

Throughout the Tokugawa Shogunate, from 1600-the mid 1800s, Japan had been in relative self-imposed isolation. Large scale trade continued with other east-asian countries, most prominently Korea and China (though the latter was, from China's perspective, illegal). The only contact with Europeans was through a small, tightly monitored trade post in Nagasaki that was granted to the Dutch. The reason for all of this was, largely, to keep out Christianity, as well as to maintain control over the various semi-autonomous realms within Japan (the Tokugawa system is complicated, and deserves its own question) by preventing them from getting outside help.

However, despite the very small amount of Western contact, there was still some knowledge of the west. This came primarily in the form of scientific and medical knowledge maintained by a group of scholars who studied 'Rangaku', or 'Dutch Studies'. The main focuses lay in things like gunnery or science that could be practical for defense or other purposes.

Ultimately, Japan was coerced into opening more to Westerners over the course of roughly a decade and a half following the arrival of US Commodore Perry in 1853. Over the next fifteen years several ports were opened to western trade. The reaction to this in Japan, though, ultimately led to the downfall of the already declining Tokugawa system and the restoration of the Emperor as the center of power (again, this is a lot of history that could go into another question). The ensuing period is known as the Meiji Restoration, and though many of its initial sentiments were reactionary and anti-western, the leaders at the time realized that the best way to combat the west was to adopt its technology and, ultimately, a lot of its culture and institutions.

Japan began a rapid period of industrialization over the next few decades, overhauling its entire social and governmental structure.

During this era, imperialism and the acquisition of colonies was seen by many as the epitome of an advanced nation, and further it would allow Japan to project its power beyond its borders. The first full on act came with the First Sino-Japanese war from 1894-95 in which Japan stunned much of the world by soundly defeating Qing Dynasty China. The result of this was the acquisition of Taiwan (then called Formosa), along with some other territories, and further it moved Korea from the Chinese sphere of influence into the Japanese.

Another major indicator of Japan's rising power came during the Boxer Rebellion in China, an uprising against rising western control. Japan was included among the 'Eight-Nation Alliance' that put down the rebellion.

Japan's first war against a European power came in 1904 when the Russo-Japanese war broke out. The impetus for the conflict was, essentially, conflicting interests around Manchuria and Korea. Japan struck first and, by the end of the war, had soundly defeated Russia by annihilating both Russia's far east and Baltic fleets, and also winning a handful of costly land battles. The end result was a massive increase in Japan's prestige, as well as some territorial gains.

Five years later Japan formally annexed Korea and would hold it for the next 35 years until the end of World War II.

After World War I, where Japan had fought with the Allies in both conquering Germany's far east colonies and also in some minor Mediterranean actions, the Washington Naval Treaty set strict guidelines on the size of various navies. Under the treaty terms, Japan would have the third-highest number of battleships in the world, after only the US and Great Britain, and furthermore their fleet was concentrated in one ocean, unlike the two larger navies.

Japan's government ultimately became more militaristic and, in the 1930s, went to war with China, first conquering Manchuria and then continuing to push inward. This and other actions led the US to cut off oil exports to Japan, a major catalyst for Japan's greater involvement in the Second World War.

During WWII, Japan for a short time controlled not only most of the Pacific, but also a large chunk of China (including major cities such as Beijing and Shanghai), and a sizeable amount of Southeast Asia stretching through Burma (now Myanmar).

So yes, Japan was a perfect example of a non-European power. IF you're interested, I'd highly recommend The Making of Modern Japan by Ken Pyle, or, for an interesting read from one of the foremost Western oriented thinkers in early-modern Japan, the autobiography of Yukichi Fukuzawa (though it really only covers the late Tokugawa era and Japan's early push into Westernization).