Is it possible to do static partial classes?

Keep naming and modifiers consistent across files:

public static partial class Facade
{
    // A few general methods that other partial facades will use
}

public static partial class Facade
{
    // Methods that are specifically for Machine Queries in our Database
}

The problem is not that the class is a partial class. The problem is that you try to derive a static class from another one. There is no point in deriving a static class because you could not make use Polymorphism and other reasons for inheritance.

If you want to define a partial class, create the class with the same name and access modifier.

you do not need to override anything, just give them the same name:

public static partial class Facade
{
    // this is the 1st part/file
}

public static partial class Facade
{
    // this is the 2nd part/file
}

You can not inherit a static class.

Static classes are sealed and therefore cannot be inherited. They cannot inherit from any class except Object.