You need a
ScrollViewer
in your control template:
<ListView.Template>
<ControlTemplate>
<Border CornerRadius="3" BorderThickness="1" BorderBrush="LightGray" Background="{TemplateBinding Background}">
<ScrollViewer Padding="{TemplateBinding Padding}" Focusable="False">
<ItemsPresenter SnapsToDevicePixels="{TemplateBinding SnapsToDevicePixels}" />
</ScrollViewer>
</Border>
</ControlTemplate>
</ListView.Template>