  Coleman's theorem states that continuous internal symmetries cannot be
spontaneously broken in two-dimensional quantum field theories (QFTs). In this
work we consider surface (i.e. two-dimensional) defects in $d$-dimensional
conformal field theories (CFTs) invariant under a continuous internal symmetry
group $G$. We study under which conditions it is possible for a surface defect
to break spontaneously a continuous internal symmetry. We find that spontaneous
symmetry breaking (SSB) is impossible under reasonable assumptions on the
defect Renormalization Group (RG) flow. Counterexamples are possible only for
exotic RG flows, that do not terminate at a fixed-point. We discuss an example
of this kind. We also illustrate our no-go result with an effective field
theory analysis of generic defect RG flows. We find a generic weakly coupled
defect universality class (with no SSB), where correlation functions decay
logarithmically. Our analysis generalizes the recent discovery by Metlitski of
the extraordinary-log boundary universality class in the $O(N)$ model.
