Quantum entanglement measures of many-body states have been ?increasingly useful to characterizephases of matter. Here we explore a ?surprising connection between symmetry-protected topology (SPT) and ?separability of their boundary mixed states. Specifically, we consider ?1d lattice systems with anomalous symmetry G, and prove that any mixed ?state ρ that is strongly symmetric under G, in the sense that Gρ~ρ, is ?necessarily tripartite non-separable. Furthermore, such states cannot ?be prepared from any tripartite separable states using finite-depth ?local quantum channels, so the non-separability is long-ranged in ?nature. The anomaly-nonseparability connection thus allows us to ?generate simple examples of mixed states with nontrivial long-ranged ?multi-partite entanglement. We further conjecture (with some evidence) ?that in general dimension d, a strongly symmetric mixed state under an ?anomalous symmetry is necessarily (d+2)-partite non-separable, in a ?long-ranged sense.