Almost five years ago, Saoirse "boats" wrote "Notes on a smaller Rust", and a year after that, revisited the idea.
The basic idea is a language that is highly inspired by Rust but doesn't have the strict constraint of being a "systems" language in the vein of C and C++; in particular, it can have a nontrivial (or "thick") runtime and doesn't need to limit itself to "zero-cost" abstractions.
What languages are being designed that fit this description? I've seen a few scripting languages written in Rust on GitHub, but none of them have been very active. I also recently learned about Hylo, which does have some ideas that I think are promising, but it seems too syntactically alien to really be a "smaller Rust."
Edit to add: I think Graydon Hoare's post about language design choices he would have preferred for Rust also sheds some light on the kind of things a hypothetical "Rust-like but not Rust" language could do differently: https://graydon2.dreamwidth.org/307291.html
Why would you want to prototype incorrect code? You don't fight with the borrow checker. The borrow checker prevents stupid mistakes. Anything that is correct that the borrow checker rejects is almost certainly a very bad idea in a prototype