Yes — the best evidence says a Greenland shark can live for about 400 years, which would make it the longest-lived vertebrate known. The age is an estimate with a wide range, not an exact birthday.
Sharks do not carry handy growth records the way trees carry rings, so researchers turned to the lens of the eye. The proteins at the center of that lens form early in life and stay put. By measuring radiocarbon in those proteins and matching it against known changes in ocean radiocarbon over time, the team could estimate when each shark was born.
That method gives a range, not a single year. For the largest shark in the study, the estimate landed around 392 years, give or take about 120 years. That is why careful writing says "about 400 years" — the shark could be somewhat younger or older than that headline number.
Almost everything about this animal is slow. It grows less than a centimeter a year in cold Arctic and North Atlantic water. Females may not be ready to have pups until they are roughly 150 years old. A shark born around the time Shakespeare was alive could, in principle, still be swimming today.
That slow pace is also a risk. When an animal takes a century and a half to start reproducing, its population cannot bounce back quickly from losses.
See the short version in the LOE PRESSURE "Things You Didn't Know" series — Episode 20, the Greenland shark.
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Sources: Nielsen et al., University of Copenhagen, eye-lens radiocarbon study of Greenland sharks, Science (2016).