The game is not currently in development.
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The Last of Us 3 or The Last of Us Part III, whichever it may be, apparently has a story already. It is by no means a final story and it may not even be fully fleshed out, but it does have a story. Be that as it may, Naughty Dog is not actually making the game right now.

Word about the non-game having a story was revealed by Naughty Dog's Neil Druckmann during an episode of the Script Apart podcast. Druckmann briefly talked about the possibility of a sequel to The Last of Us Part II. Again, keep in mind here that beyond this early story draft there is nothing going on for The Last of Us 3 right now. It could very well never happen.

"I don’t know how much I want to reveal... [co-writer Halley Gross] and I did write an outline for a story, that we're not making – but I hope one day can see the light of day – that explores a little bit what happens after this game. We’ll see.

"These games take so much to make. [Part 2], I've been thinking about it for seven years, between when the last game came out and when this came out [...] You want to make sure you’re jazzed by the idea that you have, [that] it feels like it’s challenging.

"We’ve now had two games that I feel speak to something universal as well as telling a very personal story for these characters. With one game, there’s no pattern of what a franchise is. With two games, now there’s starting to be a pattern – now I feel like there are some structural and thematic themes you'd have to stick to if you're making a third game."