The linkage of a name has nothing to do with where or how it is defined, just with where the name may be used to refer to a particular object or function.
Declaring a function inline does not force it to be inlined; it just relaxes the One Definition Rule to allow a definition in each translation unit in which it's used (and require one in each translation unit in which it's called), to make it easier to inline. It doesn't prevent a non-inline version being generated, if the compiler decides not to inline a particular call to it, or if you take the address of it.
So "external linkage" and "inline" are not exclusive; "external linkage" means that the function may be referred to in any translation unit, and "inline" means that it must be defined in any translation unit that calls it.
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