Is there a method to declare a namespace with the extern modifier so that the namespace's entire content is externally linked?
You seem to be asking two questions. One regarding external linkage and one regarding the extern
specifier.
Regarding the linkage, there is not really a need for such a syntax.
External linkage is already the default at namespace scope, whether in the global namespace or another namespace, except for const
non-template non-inline variables and members of anonymous unions. (https://en.cppreference.com/w/cpp/language/storage_duration#internal_linkage)
So the other way around, syntax to make all names in a namespace have internal linkage, is more likely to be required and this is possible with unnamed namespaces
namespace {
//...
}
in which all declared names have internal linkage.
Regarding the extern
specifier used to turn a variable definition into just a declaration at namespace scope or to give it external linkage explicitly, I am not aware of any syntax to apply it to a whole namespace. You will have to specify it explicitly on each variable.