nt.number theory - A nice variety without a smooth model

As Minhyong suggests, a curve $C$ (with $C(mathbb{Q}_p)neq0$) which has bad reduction but whose jacobian $J$ has good reduction would do the affair. This works because the cohomology of $C$ is essentially the same as that of $J$, and because an abelian $mathbb{Q}_p$-variety has good reduction if and only if its $l$-adic étale cohomology is unramified for some (and hence for every) prime $lneq p$ (Néron-Ogg-Shafarevich) or its $p$-adic étale cohomology is crystalline (Fontaine-Coleman-Iovita).

I asked Qing Liu for explicit examples. He suggested the curve
$$
y^2=(x^3+1)(x^3+ap^6)qquad (ainmathbb{Z}_p^times)
$$
when $pneq2,3$, and $y^2=(x^3+x+1)(x^3+a3^4x+b3^6)$, with $a,binmathbb{Z}_3^times$, for $p=3$.

He refers to Proposition 10.3.44 in his book for computing the stable reduction of these $C$, and to Bosch-Lütkebohmert-Raynaud,
Néron models, Chapter 9, for showing that $J$ has good reduction.

I "accept" this answer as coming from Minhyong Kim and Qing Liu.