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The answer is false due to torsion $\chi$, such as for PGL$_2$ (center trivial, hence connected!) with $\chi({\rm{diag}}(x,1)) = (-1)^{{\rm{ord}}_p(x)}$. [EDIT: This is wrong, as the OP notes below.] But if we ignore torsion, which amounts to considering continuous $\chi:T(\mathbf{Q}_p) \rightarrow \mathbf{Q}_p$, then the answer is affirmative. Indeed, this reduces [EDIT: not quite, but see comments below] to an analogous question at the level of the dual space ${\rm{Lie}}(T(\mathbf{Q}_p))^{\ast} = {\rm{X}}(T) \otimes_{\mathbf{Q}} \mathbf{Q}_p$, for which it suffices to show that the subspace of $w$-fixed points for any $w \ne 1$ is contained in some root hyperplane. This in turn is a statement of purely linear algebraic nature, so the problem over $\mathbf{Q}_p$ is equivalent to the one over $\mathbf{Q}$, which in turn is equivalent to the one over $\mathbf{R}$, where we may use considerations with Weyl chambers to conclude.