I am aware that there exist holomorphic vector bundles $E \to X$ that are smoothly trivial but not holomorphically so. For example, I think the following example works: let $X$ be a compact Riemann surface of genus $g > 0$, and let $E = \mathcal O(D)$, where $D$ is a nonprincipal divisor on $X$ of degree $0$.
However, is it possible for a holomorphic tangent bundle $TX \to X$ to be smoothly trivial, and yet have no globally nonvanishing holomorphic sections?
EDIT: Added the word “holomorphic” in the last question.