Why Does My Dock Background Sometimes Exclude the Leftmost Icon?
It turns out the Continuity feature Handoff is to blame: If I open mobile Safari on my nearby iPhone, this happens:



When I close the mobile app, the left-most segment disappears and Finder is "kicked out" of the Dock, like in the question.



Perhaps I sould file a bug report with Apple, but can not check if this has been fixed in the upcoming High Sierra or not...