You need 2 things to get this working:
1 – Add an xmlns reference in the root element of your XAML file, to the namespace where your Enum is defined:
<UserControl ...
xmlns:my="clr-namespace:YourEnumNamespace;assembly=YourAssembly">
2 – in the Value property of the DataTrigger, use the {x:Static} form:
<DataTrigger Binding="{Binding Path=LapCounterPingStatus}" Value="{x:Static my:PingStatus.PING_UNKNOWN}">
Notice that the Enum type must be prefixed with the xmlns prefix you defined above.
Edit:
If your Enum is declared inside a class you need to use the syntax:
{x:Static namespace:ClassName+EnumName.EnumValue}
for example:
{x:Static my:ConfigurationViewModel+PingStatus.PING_UNKNOWN}