Let $R$ be a dvr (whose residue characteristic is zero if it helps) with fraction field $K$. Let $A$ be an abelian variety over $K$ with good reduction over $R$. Let $X\to A$ be a finite etale morphism with $X$ geometrically connected over $K$.

Does $X$ have good reduction over $R$? (That is, does $X$ have a smooth proper model over $R$?)

By Lang-Neron, there is finite field extension $L/K$ such that $X$ is an abelian variety over $K$. In this case $X\to A$ is an isogeny and it follows from Neron-Ogg-Shafarevich that $X$ has good reduction as well over $R$. Thus, $X$ has potential good reduction over $R$, i.e., there is a finite extension $L/K$ such that $X_{R_L}$ has a smooth proper model over $R_L$, where $R_L$ is the integral closure of $R$ in $L$.

I fear that my answer has a negative answer, but I can't think of an explicit example. Can one find an elliptic curve $E$ over $\mathbb{Q}$ with good reduction at a prime $p$, an $E$-torsor $T$ with bad reduction at this prime $p$, and a finite etale cover $T\to E$. But I do not see how.