Let $C$ any curve over $k$ and $(E, e)$ be an smooth elliptic curve with distinguished point $e$. The normal bundle $\mathcal{N}_{D/X}$ with respect the effective divisor $D:= \{e\} \times C$ of the product $X:=E \times C$ is given by $$ \mathcal{N}_{D/X} = \iota_D^*(O_X(D)) $$
[cf Chap 1, 4.2 Eisenbud, Harris: "3264 and all that"] for canonical embedding $\iota_D: D \hookrightarrow X$. Even if from viewpoint of classical geometry it seems to be obvious, I would like know how I can using formal sheaf language show that in this case the normal bundle of $D$ is trivial, i.e. $\iota_D^*(O_X(D)) = O_D $.
I know that the normal bundle sits in sequence
$$ 0 \to O_X \to O_X(D) \to \iota_{D*} \iota_D^* O_X(D) \to 0 $$
and that $ \iota_D^* O_X(D)$ is a line bundle on $D$ because $O_X(D)$ is one. It has degree zero because the degree of $ \iota_D^* O_X(D)$ equals to the self intersection number of effective divisor $D$ which is zero because $D$ is the fiber of $e$ with repect to the projection onto $E$ and fibers can be moved if the base is smooth.
Therefore all I need to show $\iota_D^*(O_X(D)) = O_D $ is to check that $\iota_D^*(O_X(D))$ has non zero sections, i.e. $H^(D, \iota_D^*(O_X(D)) \neq 0$. Why that's the case?
Can it be generalized? Let $\pi: X \to C$ be a fibration (proper, flat with irreducible fibers of codimension $1$) with smooth base $C$ and $D= \pi^{-1}(c)$ an geometric fiber. Then the normal bundle $\iota_D^*(O_X(D))$ with respect to $D$ is trivial.
You can do this much more quickly. Let $p_1:X\to E$ and $p_2:X\to C$ be the projections from the product. Then noting $\mathcal{O}_X(D)\cong p_1^*\mathcal{O}_E(e)$, we see that $\mathcal{N}_{D/X} = i_D^*p_1^*(\mathcal{O}_E(e)) \cong (p_1\circ i_D)^*(\mathcal{O}_E(e))$. But the map $p_1\circ i_D$ is constant, so the pullback of any locally free sheaf of rank $r$ along this map is just $\mathcal{O}^r$.
This also gives an affirmative answer for your generalization: the composite map $D\to X\to C$ is again constant.