 Solution

Let $F$ be a $\kappa$-complete filter on $\kappa$. Use an $L_{\kappa ,\kappa }$ propositional language with a sentence $P_A$ for every $A\subseteq \kappa$. Form a theory containing $P_A$ for $A\in F$, the Boolean identitiesand, for every $\delta <\kappa$,

Every subtheory of size below $\kappa$ mentions fewer than $\kappa$ required members of $F$. Their intersection is nonempty by $\kappa$-completeness; choosing a point in it and interpreting $P_A$ as membership of that point satisfies the subtheory. The theory is therefore $\kappa$-satisfiable. Strong compactness supplies a model. Thenis an ultrafilter, contains $F$, and is $\kappa$-complete by the infinitary intersection axioms.