Consider a finite field $\mathbb{F}_q$ such that $q \equiv 1 \pmod3$ (i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}_q$ for $\omega \neq 1$) and an element $b \in (\mathbb{F}_q^*)^2 \setminus (\mathbb{F}_q^*)^3$ (a quadratic residue and cubic non-residue in $\mathbb{F}_q$). Further, I am interested in the elliptic $\mathbb{F}_q$-surface (i.e., elliptic $\mathbb{F}_q(t)$-curve) $$ E\!: \begin{cases} y_1^2 - b = b(y_0^2 - b)t^3,\\ y_2^2 - b = b^2(y_0^2 - b)t^3 \end{cases} \quad \subset \quad \mathbb{A}^4_{(y_0,y_1,y_2,t)}. $$ It is readily checked that $E$ has a Weierstrass form $W\!: y^2 = x^3 + a_4x + a_6$ with \begin{align*} a_4 \mathrel{:=}{} & -3b^4s^4 + 3b^2(b + 1)s^3 - 3(b^2 - b + 1)s^2, \\ a_6 \mathrel{:=}{} & -2b^6s^6 + 3b^4(b + 1)s^5 + 3b^2(b^2 - 4b + 1)s^4 - (2b^3 - 3b^2 - 3b + 2)s^3, \end{align*} where $s \mathrel{:=} t^3$. Note that $W$ has the infinite order $\mathbb{F}_q$-section (i.e., $\mathbb{F}_q(t)$-point) $$ x \mathrel{:=} \big(b(2bs - 1) - (3bs - 2) \big)s, \qquad y \mathrel{:=} 3\sqrt{b}(b - 1)(2\omega + 1)s^2(bs - 1). $$ Besides, the Mordell–Weil rank of $W$ over $\overline{\mathbb{F}_q}(s)$ (and hence over $\mathbb{F}_q(s)$) is equal to $1$. This immediately follows if we look at the degenerate fibers of $W$ as a rational elliptic surface. What about the Mordell–Weil rank of $W$ over $\overline{\mathbb{F}_q}(t)$ and $\mathbb{F}_q(t)$?

Let us identify $W$ with the corresponding Kodaira–Néron model, which is clearly a $K3$ surface. Again, analyzing the degenerate cases of $W$, we see that $\operatorname{rk}W\big( \mathbb{F}_q(t) \big) \leqslant \operatorname{rk}W\big( \overline{\mathbb{F}_q}(t) \big) \leqslant 5$ (even $\leqslant 3$ if $W$ is non-supersingular), because, as is known, the Picard $\overline{\mathbb{F}_q}$-number $\rho(W) \leqslant 22$ (resp., $\leqslant 20$ if $W$ is non-supersingular).

It seems that $W$ is a singular $K3$ surface (i.e., $\rho(W) \mathrel{:=} 20$) and $\operatorname{rk}W\big( \mathbb{F}_q(t) \big) = 1$. At least, for some small $q$ and $b$ (satisfying the restrictions) I checked these conjectures by means of the CAS Magma.