How can a WPF UserControl inherit a WPF UserControl?

Ensure that you have changed the first tag in the xaml to also inherit from your new basetype

So

<UserControl x:Class="TestDependencyProperty827.DataTypes.DataTypeWholeNumber"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"        
    xmlns:s="clr-namespace:System;assembly=mscorlib"
    >

becomes

<myTypes:BaseDataType x:Class="TestDependencyProperty827.DataTypes.DataTypeWholeNumber"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"        
    xmlns:s="clr-namespace:System;assembly=mscorlib"
    xmlns:myTypes="clr-namespace:TestDependencyProperty827.DataTypes"
    >

So, to summarise the complete answer including the extra details from the comments below:

• The base class should not include a xaml file. Define it in a single (non-partial) cs file and define it to inherit directly from Usercontrol.
• Ensure that the subclass inherits from the base class both in the cs code-behind file and in the first tag of the xaml (as shown above).

public partial class MooringConfigurator : MooringLineConfigurator
    {
        public MooringConfigurator()
        {
            InitializeComponent();
        }
    }



<dst:MooringLineConfigurator x:Class="Wave.Dashboards.Instruments.ConfiguratorViews.DST.MooringConfigurator"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
    xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
    xmlns:dst="clr-namespace:Wave.Dashboards.Instruments.ConfiguratorViews.DST"
    mc:Ignorable="d"
    d:DesignHeight="300" d:DesignWidth="400">

    <Grid x:Name="LayoutRoot" Background="White">

    </Grid>
</dst:MooringLineConfigurator>