Can you please share the output of echo $PATH?

Did you install Rust using pacman instead of rustup?

Can you please share the output of cat ~/.cargo/env and ls ~/.cargo/bin?

To be perfectly honest I cannot remember if I installed Rust using pacman or rustup. Is there some way to check?

echo $PATH outputs /usr/local/bin:/usr/bin:/usr/local/sbin:/var/lib/flatpak/exports/bin:/usr/lib/jvm/default/bin:/usr/bin/site_perl:/usr/bin/vendor_perl:/usr/bin/core_perl:/usr/lib/rustup/bin
cat ~/.cargo/env outputs No such file or directory
ls ~/.cargo/bin outputs mdbook rustlings

It looks like Rust was installed using pacman and it didn't add ~/.cargo/bin to your PATH environment variable. You see that rustlings is already installed in ~/.cargo/bin but it can't be found.
Either add ~/.cargo/bin manually to PATH or uninstall Rust from pacman and install it using the official way.

I will close the issue. But if you still need any help, don't hesitate to comment :)

See this section and what it says about ~/.cargo/bin in PATH. I think that Arch should just add ~/.cargo/bin to your PATH as intended instead of doing things its own way…