There are mentions of some games derived from Chess in XVII century France, but if we go down that route... why not chess? No I am not supporting this interpretation, but it could be a logical conclusion...
I think that Von Reisswitz original Kriegspiel is the first wargame that can be clerly recognized as such. As it purpose, people easily forget that it was developed not as a training tool but as a royal family pastime by von Reisswitz father and then his son pushed it more 'professional' adaptation, but it never lost the 'entertainment' side. Regimental Kriegspiel clubs flourished after Von Muffling supported the game, and these were not for formal training, but for entertainment. Moltke the elder was deeply involved in them (and this cast some strange light on certain comments by Verdy du Verois...). I played it once and it was engaging, certainly more than some commercial games I played.