Let $k$ be a field, and let $K=k(t)$ be the field of rational functions over $k$. Tate and Shafarevich [1] proved that if $k$ is a finite field, then for every $r$ there exists an elliptic curve $E/K$ with $\text{rank}~E(K)\ge r$. Their construction uses ideas from an earlier paper of Lapin [2], and indeed, the Tate-Shafarevich paper says that "analogous examples have been constructed over the field $k(t)$, when $k$ is an algebraically closed field of characteristic zero, by A. I. Lapin." However, various problems with Lapin's construction were discovered, so although the Tate-Shafarevich proof is fine for $k$ finite, the characteristic $0$ case, e.g., for $K=\mathbb C(t)$, is regarded as an open problem. (Technical note: In the case that $k$ is infinite, one requires that $E$ not be isotrivial, which means that $E$ is not isomoprhic over $K$ to an elliptic curve defined over $k$.)
Tate, J., Shafarevich, I.R., The rank of elliptic curves. Dokl. Akad. Nauk SSSR 175 (1967), 770–773.
Lapin, A.I., Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 953–988.