Others might be able to give a more in-depth accounting, as I'm by no means an aeronautical history expert, but there actually is a current Russian cosmonaut who was in space at the time of the Soviet Union's breakup.
Sergei Krikalev was aboard the Soviet space station Mir, spending more than 300 days in the cramped confines of the vessel, following a relatively ordinary launch with another cosmonaut and a British astronaut. In the chaos of the Soviet Union's disintegration, Krikalev's fate was up in the air, with the Cosmodrome in the newly-independent Kazakhstan becoming a topic of international conflict. Soviet President Mikhail Gorbachev offered the new nation a chance at sending a Kazakh astronaut into space, in a bid to secure continued use of the facility - however, this move meant that a cosmonaut with sufficient training to replace Krikalev and his skill set as a pilot would no longer be forthcoming, leaving Krikalev stranded for the time being, well past his initial five-month mission.
Krikalev was still about Mir when the Soviet Union reached its final days, with three new cosmonauts joining him. None of them, as well, possessed the necessary skill set to safely replace Krikalev aboard the station, and, less than a week later, two of these men, along with the Krikalev's original companion on this mission, back to earth. In the face of continued political unrest and a crumbling economy, Krikalev was faced with the choice to leave with the remaining cosmonaut aboard an escape capsule, or to stay aloft, fate uncertain, which would likely preserve the use of the station for future generations. With the ultimate collapse of the USSR and uncertainty over the new Russian Federation's space program, including the potential for American intervention to keep its personnel out of the hands of rogue states, Krikalev finally received word that he would be replace.
Over 5,000 trips around the planet later, Krikalev touched down in Kazakhstan, just outside the city of Arkalyk, a citizen of a country that had vanished while he was far above.
In the face of the uncertainty over Russia's future in space, the International Space Station, funded in part by the U.S. government with an eye to keeping Russia's space scientists' gazes focused skyward, was launched, the Russian-built Zarya joining with the American Unity to form the nucleus of the new station.
This unplanned extended stay would not be Krikalev's last foray outside Earth's atmosphere - far from it. Sergei Krikalev was not only the first man into the ISS, alongside former U.S. Marine aviator and current Kennedy Space Center director Robert Cabana, but has flown on the Space Shuttle and completed several subsequent tours on the International Space Station. He now trains future cosmonauts who, one can hope, have a far less eventful time in their missions aloft than he had, and find a productive future in international cooperation in the final frontier.
For contemporary coverage of Krikalev's odyssey, I recommend the LA Times' tale of Krikalev's situation, written when his fate was still uncertain, and Eric Betz's great overview article for Discover provides a better look into Krikalev's life and times.