Can I Override with derived types?

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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
    }
}

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