As confirmed by Smartsheet Support, there isn't a direct way to export a list of all archived users; the only way to get this information is either by manually scrolling through the Archived section, or by deduction (names which do not appear in the exported people list).
The existing 'People List' only includes 2 user types: Licensed Users and Managed Resources. A simple fix would be to add "Archived" as a 3rd user type.