Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ${\bf F}_p$ (where $p>0$ is a prime number).

My question is : does there exist an abelian variety $A$ over $K$, with the following properties :

(a) the $K|k$-image of $A$ is trivial;

(b) there exists an étale $K$-endomorphism of $A$, whose degree is a power of $p$ ?

The condition on the $K|k$-image can be rephrased as : there are no non-zero $K$-homomorphisms

$A\to C_K$, where $C$ is an abelian variety over $k$.

For examples of abelian varieties satisfying condition (b) only, look at abelian varieties $C_K$, where $C$ is an ordinary abelian variety over ${\bf F}_p$. The abelian variety $C$ is endowed with the étale endomorphism given by the Verschiebung morphism.

Also, notice that if there is an abelian variety over $K$ satisfying (a) and (b), then the dimension of $A$ is larger than one (ie it is not an elliptic curve). Indeed, if an elliptic curve $E$ satisfies the above conditions, then $E$ has an endomorphism, which is not a multiplication by a scalar and thus it has complex multiplications; this implies that it is isogenous to an elliptic curve defined over $k$, by a theorem of Grothendieck (or by more direct arguments).

Finally, I would like to point out that if $A$ is an ordinary abelian variety over $K$, which has maximal Kodaira-Spencer rank, then $A[p]$$(K^{\rm sep})=0$ by a theorem of J-F Voloch (see p. 1093 in "Diophantine Approximation on Abelian Varieties in Characteristic $p$", Amer. J. Math., Vol. 117, No. 4., pp. 1089-1095); this shows that such an abelian variety cannot have an endomorphism as in (b).