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In the 1st part of this work [DHdS18], we studied affine group schemes over adiscrete valuation ring (DVR) by means of Neron blowups. We also showed how toapply these findings to throw light on the group schemes coming from Tannakiancategories of D-modules. In the present work, we follow up this theme. We showthat a certain class of affine group schemes of "infinite type", Neron blowupsof formal subgroups, are quite typical. We also explain how these group schemesappear naturally in Tannakian categories of D-modules. To conclude, we isolatea Tannakian property of affine group schemes, named prudence, which allows oneto verify if the underlying ring of functions is a free module over the basering. This is then successfully applied to obtain a general result on thestructure of differential Galois groups over complete DVRs.

On the structure of affine flat group schemes over discrete valuation rings, II