I'm assuming that when you say "implementation of Vulkan in C#", you are referring to bindings that let you call into Vulkan. I maintain one such set of bindings for .NET: https://github.com/mellinoe/vk. I've built it mainly for myself and for my internal use in my abstract graphics library Veldrid, where it provides the FFI for my Vulkan backend. As such, it's pretty specific to my needs. It is an extremely "raw" set of unsafe bindings. There's no fancy wrapping or marshalling happening, and its usage is intended to be identical to Vulkan in other languages. It does not support Vulkan 1.1 as of yet, but I haven't seen other bindings support that yet, either.
There's a couple of other options that I've seen, which might be more focused on "public consumption" than mine. https://github.com/discosultan/VulkanCore looks good, and is a higher-level set of bindings, with more of an intermediate layer between you and native Vulkan. https://github.com/FacticiusVir/SharpVk is another that I've seen used. I don't have experience with using either of these, personally.