I've seen both Rust's lifetimes/borrow-checker and Objective-C/Swift's automatic reference counting described as "zero cost" (since the bulk of the work is done at compile time).
Rust's borrow checker is zero-cost because it's not doing runtime analysis. But it's not a garbage collector, unless you're using Steve Klabnik's "static garbage collector" terminology [1]. Reference counting is definitely not zero cost. It reduces GC runtime latency for most workloads, but not to zero, and it does so at the cost of reduced bandwidth.
1: https://words.steveklabnik.com/borrow-checking-escape-analys...