Short answer: yes, and in some cases they are even designed to.
Longer answer: Lady Lumiya, an Extended Universe character, was trained as a Sith by Darth Vader somewhere in the general timeline of ESB/RotJ (violating the Rule of Two, but this rule was commonly broken by both Palpatine and Vader during their time). She was on a pilgrimage to the ancient Sith world Ziost when the Death Star was destroyed, and as such became the last living Sith after the Empire's defeat. From Darth Vader, she had acquired a shard of Kaiburr crystal, a very powerful Force-focusing material, and used it to build a highly flexible lightsaber weapon called a "lightwhip".
Lumiya's weapon, though rare, was not unique; lightwhips are seen in several forms in the hands of Force practitioners through the Extended Universe history, mostly in the hands of Sith, Nightsisters and other Dark Side Force users (though Obi-Wan Kenobi is a notable exception, having fashioned and used a lightwhip for a time during the Clone Wars).
Lightwhips were generally used not for any great power inherent in the design, but because they were rare and thus unfamiliar to one's opponent; in a battle to the death against a weapon you've never seen before, you learn quickly or die.
Below is Sith Lady Githany, the first known lightwhip user around 1000BBY.

Lumiya's lightwhip is similar in theory, but has multiple "strands", to form a cat-o-nine-tails-style weapon:
