Is it possible to write a story with a single character and no development of that character at all?
You can always have a character who doesn't develop; flat Disney villains come to mind. But the flat character is generally in opposition to the hero/ine, who does develop.
So the question is, why would you write such a story? What could possibly happen in it? If you have one character, period, and that character doesn't develop, what is that person doing?
The TV show "Seinfeld" is an example of a show where the characters didn't develop. They never learn anything about themselves and this was a source of humor in the show. Or at least it must have been for the people who liked it (and there was a lot of them), personally I never really got into it.
I think also some of Samuel Beckett's work would have one or two characters and little or no character development, some of it is also comic, and indeed absurd.
Is it possible? Probably. It may depend on the definition of "character development". I was just looking for a definition and didn't find one in 30 seconds, but it's normally understood to mean (a) revealing the nature of a character to the reader, and/or (b) a character growing and changing over the course of the story.
By definition (a), if you write, "Bob walked into the room", you have told us several things about the character: He is named "Bob". He is capable of walking. He has some reason for wanting to be in this room. Okay, trivial things, but you have begun the process of character development. It's hard to imagine a story where you literally tell the reader absolutely nothing about any of the characters, even indirectly.
It's a little more plausible by definition (b). I suppose you could imagine a story where the characters learn nothing and do not change in any way. But at the very least, you would think that SOMETHING must happen to the characters in the course of the story, so if nothing else they have gained experience. Even if you do not spell out how the character's react to these experiences, the reader is likely to draw inferences.
Perhaps you could be more specific about what you have in mind.
It definitely is possible. If you don't want them to grow, then you should probably have another character who does grow. A reader finds a character growing from one extreme to another interesting.
If a character does not grow at all, you may want to make them close to, but not perfect. If it is the protagonist who does not grow, you want them to be at least a little noble, honorable, and possibly even have a past where did grow, but now are done growing. If it is the antagonist, then you should probably make them mean, malicious, and all the key elements of an antagonist.
Writing a story is a journey in itself, but what your characters do is up to you. If they don't grow at all, I would suggest you make sure their life is filled with challenges so that way it still is interesting to see how they handle each challenge.
It is possible. How, I know not, but it is possible. I once wrote a story that was literally a narration of events with no character, and the community (it was a fan fiction) really liked it. I still don't know how that happened.
A character did appear later on, but there was no character development. The closest it got was when the character almost sacrificed himself so someone could escape, but that event had no impact on anything else whatsoever. The character never changed.
Needless to say, that was written before I knew anything about development.
Normally, when the main character remains static, that character is actually a catalyst character and the world or characters around them are the true protagonists. A popular example is Ferris Bueller's Day Off. Ferris Bueller may be the titular character, but it is more or less agreed that his best friend Cameron is the real protagonist, Ferris merely gives Cameron an impetus to change.
This may be an interesting route to try if you were to write a story with a single unchanging character!
Some development must occur to have a successful story, even if it's a change in the audience's understanding rather than one inside of the story itself. Otherwise, there would be no point in consuming it, it'd be the literary equivalent of empty calories.
Anything is possible in writing. There may be quality issues though. Wait, let me get this straight. You want to write a single story with one static character total? You could, but I can only think of one other story that a single character and thats Hatchet by Gary Paulsen and that was a dynamic character.
I am actually adding a second answer based on something @user16583 mentioned.
In some long-running comic strips, characters don't age or change. Strips like FoxTrot and Sally Forth occasionally make sly meta-jokes referencing the idea that the kids in the strip have been 10 or 12 for decades. Beetle Bailey has been doing the same thing for 70 years.
But even really long-running strips have some changes: The cast of Peanuts expanded, Blondie has gotten a job, there have been new platoon members and sexual harassment training at Camp Swampy.
So while you might be able to pull this off with a series of short stories (like Jeeves & Wooster, perhaps) or a graphic novel, particularly if the stories are meant to be funny, I think you'd be hard-pressed to make it work with drama.
This summary of K M Weiland's Creating Character Arcs: The Masterful Author's Guide to Uniting Story Structure, Plot, and Character Development discusses examples of her observation that the protagonist's character development often involves their discovering they believed a big lie, and is avoided if they consistently believe a big truth and can influence others to adopt it, even unintentionally. The two Paddington films are universally critically acclaimed examples of the latter approach.
An author worried their protagonist doesn't develop should review how others consequently develop around them, and decide whether (and if so how) they want to change the story so the protagonist develops as well. I've written at least one work in which, while the protagonist's influence in their world is largely characterized by how others develop because of them, they develop as well but in a dissimilar way.