I’m not sure about this, but I think it’s some sort of tree!
After looking into it for a little bit, it does look like there is a pine tree in North America that is just over 5000 years old, and is the oldest plant on earth. However, researchers have found remains of deep sea sponges that were over 10,000 years old, so there might well be some of these primitive animals hidden in the deep sea still alive.
More excitingly, (especially to myself), there is also some extremely long-lived microbial life. Again in the deep sea, endoliths (microorganisms that live in microscopic rock cracks) are extremely slowly metabolising organisms (they double once every 10000 years or so, and grow very, very slowly) and might even live for millions of years!