manifolds, following several partial results (cf. [@Bu] and the references in [@BeVe; @BeVe2]), an inhomogeneous transference principle was developed by Beresnevich and Velani using which they resolved the inhomogeneous analogue of the Baker-Sprindžuk conjectures. Subsequently, Badziahin, Beresnevich and Velani [@BaBeVe] established the convergence and divergence cases of the inhomogeneous Khintchine theorem for nondegenerate manifolds. They proved a new result even in the classical setting by allowing the inhomogeneous term to vary. The divergence theorem is established in the same paper in the more general setting of Hausdorff measures.

In this paper, we will establish the convergence case of an inhomogeneous Khintchine theorem for nondegenerate manifolds in the $S$-adic setting, as well as the divergence case for ${\mathbb{Q}}_p$. As in [@BaBeVe], the divergence case is proved in the greater generality of Hausdorff measures. Prior results in the $p$-adic theory of inhomogeneous approximation for manifolds focussed mainly on curves, cf. [@BDY; @BeK; @U1; @U2].

Main Results
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To state our main results, we introduce some notation following [@MoS1], recall some of the assumptions from that paper and set forth one further standing assumption. The assumptions are as follows.

1.  $S$ contains the infinite place.

2.  We will consider the domain to be 