Any modification of the theorem where the definition of "cover" you give is local on the base and contains inverse limits of finite etale covers (e.g. flat plus unramified as in the original question) will also be false because the property of being an inverse limit of finite etale covers is not local on the base.
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
