Currently our team deletes the items altogether, otherwise they clutter the Kanban board, obscuring the currently relevant work items. Is there an option to archive an item, so it disappears from the Kanban board completely?
Kanban Boards (also called Team boards) in Visual Studio team services are driven by Area and Iteration Fields. You can create a new iteration say "Archive" and move the closed work items in to that path, they will not show up on the board and will be available for future reference.
Archiving work items is done at Database level and it is more expensive operation.