As Minhyong suggests, a curve (with ) which has bad reduction but whose jacobian has good reduction would do the affair. This works because the cohomology of is essentially the same as that of , and because an abelian -variety has good reduction if and only if its -adic étale cohomology is unramified for some (and hence for every) prime (Néron-Ogg-Shafarevich) or its -adic étale cohomology is crystalline (Fontaine-Coleman-Iovita).
I asked Qing Liu for explicit examples. He suggested the curve
when , and , with , for .
He refers to Proposition 10.3.44 in his book for computing the stable reduction of these , and to Bosch-Lütkebohmert-Raynaud,
Néron models, Chapter 9, for showing that has good reduction.
I "accept" this answer as coming from Minhyong Kim and Qing Liu.
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