Is there a fully functional Selenium WebDriver for MS Edge?

They released a version from the Windows 10 fall update that can be found here: https://www.microsoft.com/en-us/download/details.aspx?id=49962

Status of the WebDriver development can be found here: https://dev.windows.com/en-us/microsoft-edge/platform/status/webdriver/details/

So that was really the solution (thank you, Martin Beeby!): If you want to use XPath locators, then you have to use the latest Microsoft Edge WebDriver (2.0 at the time of writing), even though http://seleniumhq.org still reference 1.0 on their download page. You can get it here:
https://www.microsoft.com/en-us/download/details.aspx?id=49962

But if you want to use the latest driver, then you have to use the latest Edge version. But if you want to use the latest Edge version, you have to update your Windows 10 from "Build 10240" to "Version 1511" aka "Fall Update", formerly known as "Threshold 2".

You are said to get this update via WindowsUpdate, but not before you've been using your Windows 10 for at least 31 days already. Of course I cannot confirm this, as I didn't want to wait for an entire month. But Microsoft themselves say so:
http://windows.microsoft.com/de-de/windows-10/windows-update-faq

As a workaround for immediate update, you are said to be able to update manually via Microsoft's "Media Creation Tool", which I tried, but it didn't work. Instructions here:http://www.groovypost.com/howto/install-windows-10-november-update-1511-manually-media-creation-tool/

If you're an MSDN member (like me, BTW), this version of Windows shows up as a separate product: You have one product named "Windows 10" and another product named "Windows 10, Version 1511" available for download. The latter one is the one I'm using now, and my test cases are running. Even though all of them are still red, but that's a different subject. ;-)