No, you don't have data races in Rust. Deadlocks are more or less just as possible in Rust as in other languages. In fact, the 2024 edition of Rust changed something[0] to make it more difficult to hit some deadlock situations, but it's well-known that there's not much in the language to protect you from them.

[0] https://doc.rust-lang.org/edition-guide/rust-2024/temporary-...