Can I Override with derived types?

You can re-declare (new), but you can't re-declare and override at the same time (with the same name). One option is to use a protected method to hide the detail - this allows both polymorphism and hiding at the same time:

public class Father
{
    public Father SomePropertyName
    {
        get {
            return SomePropertyImpl();
        }
    }
    protected virtual Father SomePropertyImpl()
    {
        // base-class version
    }
}

public class Child : Father
{
    public new Child SomePropertyName
    {
        get
        { // since we know our local SomePropertyImpl actually returns a Child
            return (Child)SomePropertyImpl();
        }
    }
    protected override Father SomePropertyImpl()
    {
        // do something different, might return a Child
        // but typed as Father for the return
    }
}

This is not possible in any .NET language because of type-safety concerns. In type-safe languages, you must provide covariance for return values, and contravariance for parameters. Take this code:

class B {
    S Get();
    Set(S);
}
class D : B {
    T Get();
    Set(T);
}

For the Get methods, covariance means that T must either be S or a type derived from S. Otherwise, if you had a reference to an object of type D stored in a variable typed B, when you called B.Get() you wouldn't get an object representable as an S back -- breaking the type system.

For the Set methods, contravariance means that T must either be S or a type that S derives from. Otherwise, if you had a reference to an object of type D stored in a variable typed B, when you called B.Set(X), where X was of type S but not of type T, D::Set(T) would get an object of a type it did not expect.

In C#, there was a conscious decision to disallow changing the type when overloading properties, even when they have only one of the getter/setter pair, because it would otherwise have very inconsistent behavior ("You mean, I can change the type on the one with a getter, but not one with both a getter and setter? Why not?!?" -- Anonymous Alternate Universe Newbie).