To see this, proceed similarly to Scott Carnahan's example, but instead of gluing a chain of P^1's $\mathbb{P}^1$'s together, glue together P^1's $\mathbb{P}^1$'s "indexed by Spec C[...,x_{-1},x_0,x_1,...]/.$\operatorname{Spec} \mathbb{C}[...,x_{-1},x_0,x_1,...]/\langle x_i^2-1\rangle$." Explicitly, let the base curve B $B$ be two P^1's $\mathbb{P}^1$'s glued together at two distinct points. Over each P^1 $\mathbb{P}^1$ consider the affine morphism corresponding to the sheaf of algebras calO(P^1)[...,x_{-1},x_0,x_1,...]/$\mathcal{O}_{\mathbb{P}^1}[...,x_{-1},x_0,x_1,...]/\langle x_i^2-1\rangle$. At one of the points, glue together the two possible x_i's. $x_i$'s. At the other, glue x_i $x_i$ to x_{i-1}. $x_{i-1}$. Over each P^1, $\mathbb{P}^1$, the resulting morphism is an inverse limit of finite covers, but over all of B, $B$, it is not. This is written down fully in Warning 2.5b of http://math.harvard.edu/~kwickelg/papers/VW.pdf -- Kirsten Wickelgren