  In this work, we consider $M=(\mathbb{B}^3_r,\bar{g})$ as the Euclidean
three-ball with radius $r$ equipped with the metric $\bar{g}=e^{2h}\left\langle
, \right\rangle$ conformal to the Euclidean metric. We show that if a free
boundary CMC surface $\Sigma$ in $M$ satisfies a pinching condition on the
length of the traceless second fundamental tensor which involves the support
function of $\Sigma$, the positional conformal vector field $\vec{x}$ and its
potential function $\sigma,$ then either $\Sigma$ is a disk or $\Sigma$ is an
annulus rotationally symmetric. In a particular case, we construct an example
of minimal surface with strictly convex boundary in $M$, when $M$ is the
Gaussian space, that illustrate our results. These results extend to the CMC
case and to many others different conformally Euclidean spaces the main result
obtained by Haizhong Li and Changwei Xiong.
