You would still have earned a black belt in Karate, yes. In the same way that somebody who has earned a law degree, but never practices law and it way out of date on current laws still has an LLB. Black belts are not necessarily the end point. A black belt really kind of means you've mastered all the basic techniques and can now train people, like it's basically like "you're not an apprentice any more", so to people really devoted to Karate or similar martial arts, it's often like a start point more than an end point. Like having a degree, once you have your BA/BSc or similar, you move onto things like an MA/MSc and then a PhD, and in those postgrad courses, your learning and research is more self-directed and then you present it for others to learn from. You're not being taught things directly so much as honing your craft or using your knowledge to do research. A black belt is like a BA or BSc, representing the end of being directly taught the library of moves and sequences by a master.
I have a green belt in karate from when I was in my teens. I'm not quite so flexible as I was twenty years ago and haven't done karate in years, so if I went back into it, I'd need to work hard to remember a lot of the technical stuff, like katas and some of the more unusual techniques and precise stances and footwork. I wouldn't lose my green belt, but I'd need to remember all the stuff up to green belt before I could start studying towards blue, brown and then black.
But I will say this: If somebody got up to a black belt, no matter how old and out of shape they are, they will probably be pretty dangerous in an actual fight against anyone who isn't well-trained in martial arts. Because the thing is, yeah, I've forgotten most of the katas (sequences of moves for learning, grading and demonstration purposes) and I can't do head-height kicks or jump kicks so well as I could as a seventeen year old, but none of those are useful in a real fight for the most part anyway. The stuff that stays with you because it's so drilled in, like how and where to punch effectively, how to do a really killer kick to the knee or an elbow strike- that stuff stays with you and it's reflexive because you drill it over and over and over, and generally so long as you have at least a modest amount of strength and mobility left, you'll be pretty dangerous. If I punch, even 20 years later, it'll still be a Karate punch, it's just so drilled into my muscle memory, and I'm just a green belt!
And if they have kept training, even if they are old and fat, they might be a total badass. I had this one sensei, a black belt, who was this big old scottish guy. White hair and beard, big gut, proper old viking looking fellow with a Glaswegian accent (He'd say stuff like "Ach, Katie, ye cannae punch people in a real fight! Ye'll break yer wee chickenbone hands! Yeh should gi' 'em a good kickin' wi' yer feet!"), could still jump into the air and do a split-kick hitting two pads either side of him at shoulder-height. It was incredible. He was SO FAST in a fight, you wouldn't even believe it.
