2 added 975 characters in body; edited body

Edit: So as not to drag things out, I hope it's okay if I just give you a standard example. Let

$E_0: y^2 = x^3 + Ax + B$

be your favorite elliptic curve over $\mathbb{Q}$ (i.e., any will do). Consider the elliptic curve $E: t y^2 = x^3 + Ax + B$
over the rational function field $\mathbb{Q}(t) = \mathbb{Q}(\mathbb{P}^1)$. Spreading this out as a scheme over $\mathbb{P}^1_{/\mathbb{Q}}$, we see that there are two singular fibers, at $t = 0$ and $t = \infty$. Discarding these we get an elliptic curve over $\mathbb{A}^1 \setminus \{0\}$ which is isotrivial -- the $j$-invariant over every fiber is $j(E_0)$ -- but nontrivial: the isomorphism classes of the fibers are in bijection with $H^1(\mathbb{Q},\mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$.

It's a good bet that you'll find this example somewhere in the chapter on elliptic surfaces in Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.

Can it be done over any base scheme? Unless I misunderstand, of course not, e.g. not over the spectrum of a field.

1