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In characteristic $0$, I think the answer is yes, since $U(k)$ contains elements with square zero. See Theorem 11.1 in Brendan Hassett, Potential density of rational points on algebraic varieties (pdf). It says that a K3 surface $X$ over a number field possesses an elliptic fibration iff there exists $D \in \operatorname{NS}(X)$ such that $D^2 = 0$. It seems to me I see no reason that the proof goes through for should not work over any field of characteristic zero, but of course this has to be checked.
In characteristic $0$, I think the answer is yes, since $U(k)$ contains elements with square zero. See Theorem 11.1 in Brendan Hassett, Potential density of rational points on algebraic varieties (pdf). It says that a K3 surface $X$ over a number field possesses an elliptic fibration iff there exists $D \in \operatorname{NS}(X)$ such that $D^2 = 0$. It seems to me that the proof goes through for any field of characteristic zero.