Let \(K=k((t))\) be a field of formal power series with coefficients in a perfect field \(k\) of characteristic \(p\neq0\). Let \(A\) be an abelian scheme defined over the ring of integers \(R\) in \(K\) with the dual abelian scheme defined over the ring of integers \(R\) in \(K\) with the dual abelian scheme \(\hat A\) The main result of the paper states that there exists a non-degenerate pairing, \(H^1(\bar K, a\otimes_R\bar K)\times\pi_1(\bar A)\to\mathbb Q/\mathbb Z\), which is a perfect duality of topological \(G\)-modules, \(G\) being the Galois group of \(k\). Here \(\bar K\) denotes the maximal unramified extension of \(K\) with the ring of integers \(\bar R\), and \(\pi_1(\bar A)\) is the fundamental group (in the sense of Serre) of the proalgebraic group associated to \(\bar A=A\otimes_R\bar R\). The theorem generalizes - in the equicharacteristic case - the theorem of Ogg and Šafarevič [cf. I. R. Shafarevich, Transl., Ser. 2, Am. Math. Soc. 37, 85–114 (1964); translation from Tr. Mat. Inst. Steklov. 64, 316–346 (1961; Zbl 0142.18401)] to the \(p\)-primary components of the groups in question. In order to obtain such a pairing, the author first proves a duality theorem in flat cohomology for finite flat commutative group schemes over \(R\), using some exact sequences recently discovered by M. Artin and J. Milne [Invent. Math. 35, 111–129 (1976; Zbl 0342.14007)], and then reads off the result from the sequence of multiplication by \(p^n\) on \(A\).