Arithmetic differential operators on a semistable model of $${\mathbb {P}}^1$$ P 1

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In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also consider a refinement of the order filtration on the sheaf of level zero (before taking the p-adic completion). The associated graded sheaf, which we explicitly determine, explains to some extent the occurrence of the cohomology classes in degree one.