Since nobody else has mentioned this, let me point out the article of Bogomolov and Tschinkel.

MR2154371 (2006e:14024) Reviewed <br>
Bogomolov, Fedor(1-NY-X); Tschinkel, Yuri(D-GTN) <br>
Rational curves and points on K3 surfaces. (English summary) <br> 
Amer. J. Math. 127 (2005), no. 4, 825–835. <br>
14G05 (11G35 14J28) <br>
[http://arxiv.org/pdf/math/0310254.pdf](http://arxiv.org/pdf/math/0310254.pdf)

Let $Z$ be a Kummer K3 surface with a specified very ample divisor class $D$, let $Y$ be $\mathbb{P}^1$ with one marked point $0$, and let $X$ be $Y\times Z$ with its obvious projection $\text{pr}_Y$ to $Y$, with the $Y$-ample divisor class $\text{pr}_Z^*D$, and with an arbitrary $K$-rational point $x_0$ in the fiber over $0$ marked.  Then Bogomolov and Tschinkel prove that for every finite field reduction $R/\mathfrak{m}$, there exists a section $s_{\mathfrak{m}}$ sending $0$ to $x_0$ and having positive degree with respect to $\text{pr}_Z^* D$, i.e., the section is not "horizontal".  They point out that <b>it may well be</b> that the same holds over $\overline{K}$, but nobody really knows this, and some people are skeptical.  So this is an example worth bearing in mind.