I can add scrolling over the tab items of a TabControl using information from this post: https://github.com/AvaloniaUI/Avalonia/discussions/11896

However, a rather unfortunate side-effect is that the contents of the tab items disappear (in my sample xaml below, only tab item ‘One’ has contents).

Here is simplified xaml which demonstrates the problem:

<TabControl Width="200">
    
    <!-- this allows scrolling through the tab items, but hides the tab item contents! --> 
    <TabControl.Styles>
        <Style Selector="TabControl">
            <Setter Property="Template">
                <ControlTemplate>
                    <ScrollViewer HorizontalScrollBarVisibility="Auto">
                        <ItemsPresenter ItemsPanel="{TemplateBinding ItemsPanel}"/>
                    </ScrollViewer>
                </ControlTemplate>
            </Setter>
        </Style>
    </TabControl.Styles>
            
    <TabItem Header="One">
        <DockPanel>
            <CheckBox Content="First option"/>
            <CheckBox Content="Second option"/>
        </DockPanel>
    </TabItem>

    <TabItem Header="Two">
    </TabItem>

    <TabItem Header="Three">
    </TabItem>

    <TabItem Header="Four">
    </TabItem>

    <TabItem Header="Five">
    </TabItem>

    <TabItem Header="Six">
    </TabItem>
</TabControl>