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A 2- (v , k , λ) block design (P , B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G , +) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G , the embedding of P in G is also such that, conversely, any zero-sum k -subset of P is a block in B , then (P , B) is said to be strongly additive. In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2- (16 , 4 , 2) design (F 4 × F 4 , B 2) , which decomposes into two disjoint isomorphic copies of the affine plane of order four. An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG (4 , 2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem. [ABSTRACT FROM AUTHOR]