The System.Diagnostics assembly is part of the Silverlight CLR framework, but it only includes classes related to debugging, the Process class is not available. Is there any other way to start an outside process from a Silverlight application?
Yes, you can in Silverlight 4 (in out-of-browser with elevated full trust), example:
dynamic cmd = AutomationFactory.CreateObject("WScript.Shell");
cmd.Run("calc.exe", 1, true);