Sort of. In 1954 in Macau there was a kungfu competition between Wu Gongyi (53) and Chen Kefu (35), two local martial arts experts. Wu Gongyi had put out an ad challenging all-comers, and Chen Kefu accepted it.

Video of the fight can be seen here: https://www.youtube.com/watch?v=2FsZyPjsjTA

Be warned though, you won't be impressed with that fight. But it was a big deal in Hong Kong and Macau when it happened, and led to the revival of the wuxia genre of fiction (basically, martial arts fiction) in Hong Kong, as newspapers began serializing wuxia novels.

In the post-war streets of Hong Kong, there were young, otherwise idle, martial arts masters from various schools. Being idle young men, they found diversions and the illegal, bare knuckle street fighting circuit known as 'beimo' was founded.

Among the most famous practitioners of this was Wong Shun Leung, student of the legendary Yip Man. Wong allegedly won 60 fights with the then unknown Wing Chun style. Wong rapidly became a sort of underground celebrity and helped to spread the popularity of Wing Chun before one Bruce Lee exploded on to the scene.

Not sure if this constitutes "epic," but Bruce Lee did have a pretty publicized fight with another martial artist, Wong Jack Man. The fight was private, so accounts vary, but Lee and his side claim that Wong Jack Man tried to run away, then Lee pounced on him and hit him repeatedly on the back of the head. So, that's an example, but I think it's useful to show that martial arts duels most likely did happen, at least on a small scale.

If you're talking about a real, unsanctioned street fight between Karate or Kung Fu practitioners then plenty of underground challenge matches have occurred throughout history. For instance Bruce Lee fought a Kung Fu master named Wong Jack Man in a private duel which was publicized in Chinese newspapers in San Francisco's Chinatown. If you're talking about sanctioned fights then I would follow the careers of Cung Le or Lyoto Machida. Cung Le was undefeated in a Chinese combat sport called San Shou and has been successful in Mixed Martial Arts. Lyoto Machida has also been very successful in Mixed Martial Arts. Both of them fight in the UFC. You're unlikely to find video footage of a Martial Arts fight that rivals what you see in the movies because movie fights are choreographed. Actors can do all kinds of flashy moves and display ridiculous speed and timing as well as exaggerate power. If you want to see epic fights watch MMA.