Geocore [Steam], an Early Access six degrees of freedom shooter has Linux support and it was recently updated to include Vulkan too. Before getting excited, keep in mind this it's £1.99, so don't expect much from it.
It's actually using Unity, so it's not surprising they were able to build it with Vulkan. I've tested it briefly and it seems the Vulkan version does work on Linux, although the game is locked to a max of 60FPS so you can't tell easily if the performance is any better.
It has the same fullscreen bug as Rust, where you get no input so you have to load it up in Windowed mode. I assume this is a Unity engine bug, since multiple titles have it. If you make it fullscreen once it's loaded, it then works fine.
I tested it out, but it's incredibly rough. It doesn't really have any players online, but it could be one to keep an eye on. If they build out the single-player mode it could be a reasonable six degrees of freedom shooter in future. Going by their updates on Steam, they do seem to be working on a proper single-player mode for it.
For now, in my opinion it's not worth picking up as there's just nothing to it.