= Solution
A uniform example is $\boxed{A=\kappa\setminus\{\omega\}}$. It contains the final segment $[\omega+1,\kappa)$, a <club set>. Therefore $A$ belongs to the <club filter>. Every two <club sets> in a regular uncountable cardinal intersect: alternately choose larger points from each and take the countable supremum, which remains below $\kappa$ and lies in both by closure. Hence every member of the <club filter> is a <stationary set>, so $A$ is stationary.
However, every finite <ordinal> belongs to $A$, and their increasing supremum $\omega$ does not. Thus $A$ is not closed and is not a <club set>. This is a <stationary nonclosed member of the club filter> for every regular uncountable $\kappa$.
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