How about Vatican City? It's a non-secular state existing within the boundaries of Italy and ruled by the Pope. A large number of the inhabitants are priests, nuns, and other religious folks.

OFFICIALLY, England is ruled by the head of the Anglican Church, the Queen (or King). But I wouldn't classify it as non-secular.

Japan, likewise, is ruled by the Emperor, who is the head of the church and is traditionaly considered divine. Again, I wouldn't say it's not a secular society.

Traditional Tibet? The Pharaohs of ancient Egypt, like the Roman Emperors, were considered gods, but were they secular societies or not?

China today is officially atheist, as was the old Soviet Union. Can a society be so militantly atheistic that it moves from secular to non-secular -- that it makes atheism its religion, so to speak?

The US is officially secular, except for the current president....

I don't think that your answer can be as clear-cut as you'd like it to be.