Update: The WebSocket library is available as a fully supported component at https://www.rebex.net/websocket/.
System.Net.WebSockets namespace is already available on Xamarin, although it doesn't support TLS 1.2 on all platforms.
However, we implemented a simple multiplatform WebSocket library that uses TLS/SSL and HTTP API provided by
Rebex.Networking assembly. We are considering whether to release it to the public.
This means that our implementation support TLS 1.2 on all platforms, including Xamarin.Android/Xamarin.iOS, .NET 2.0/3.5, .NET CF 3.5/3.9 and .NET 4.0 on Windows 7 or earlier (where System.Net.WebSockets is not implemented).