  It was recently shown by R. Souam and E. Toubiana that the (non constantly
curved) Berger spheres do not contain totally umbilic surfaces. Nevertheless in
this article we show, by perturbative arguments, that all analytic metrics
sufficiently close to the round metric $g_{0}$ on $\mathbb{S}^{3}$ possess
\textsl{generalized} totally umbilic 2-spheres, namely critical points of the
conformal Willmore functional $\int_{\Sigma}|A^\circ|^{2}\,d\mu_{\gamma}$. The
same is true in the smooth setting provided a suitable non-degeneracy condition
on the traceless Ricci tensor holds. The proof involves a gluing process of two
different finite-dimensional reduction schemes, a sharp asymptotic analysis of
the functional on perturbed umbilic spheres of small radius and a quantitative
Schur-type Lemma in order to treat the cases when the traceless Ricci tensor of
the perturbation is degenerate but not identically zero. For left-invariant
metrics on $SU(2)\cong \mathbb{S}^{3}$ our result implies the existence of
uncountably many distinct Willmore spheres.
