We consider linear maps T : X → Y, where X and Y are polar local convex spaces over a complete non-archimedean field K. Recall that X is called polarly barrelled, if each weakly * bounded subset in the dual X is equicontinuous. If in this definition weakly * bounded subset is replaced by weakly * bounded sequence or sequence weakly * converging to zero, then X is said to be ∞ -barrelled or c 0 -barrelled, respectively. For each of these classes of locally convex spaces (as well as the class of

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... paces with weakly * sequentially complete dual) as domain class, the maximum class of range spaces for a closed graph theorem to hold is characterized. As consequences of these results, we obtain non-archimedean versions of some classical closed graph theorems. The final section deals with the necessity of the above-named barrelledness-like properties in closed graph theorems. Among others, counterparts of the classical theorems of Mahowald and Kalton are proved.