An open question posed by Fawzi and Ferme [Transactions on Information Theory 2024], asks whether non-signaling (NS) assistance can increase the capacity of a broadcast channel (BC). We answer this question in the affirmative, by showing that for a certain K-receiver BC setting, called Coordinated Multipoint (CoMP) that arises naturally in wireless networks, NS-assistance provides multiplicative gains in capacity and degrees of freedom (DoF), even achieving K-fold improvements in some cases. Somewhat surprisingly, this is shown to be true even for 2-receiver broadcast channels that are semi-deterministic and/or degraded. In a CoMP BC, B single-antenna transmitters, supported by a backhaul that allows them to share data, act as one B-antenna transmitter, to send independent messages to K receivers, each equipped with a single receive antenna. A fixed and globally known connectivity matrix M, specifies for each transmit antenna, the subset of receivers that are connected to (have a non-zero channel coefficient to) that antenna. Besides the connectivity, there is no channel state information at the transmitter. The DoF region is fully characterized for a class of connectivity patterns associated with tree graphs. Sum-capacity with NS-assistance for arbitrary connectivity patterns is bounded below and above by the triangle number and the min-rank of the connectivity matrix, respectively. While translations to Gaussian settings are demonstrated, most of our results are presented under noise-free, finite-field (Fq) models. Converse proofs for classical DoF adapt the Aligned Images bounds to the finite field model. Converse bounds for NS-assisted capacity extend the same-marginals property to the BC with NS-assistance available to all parties. Even stronger (unbounded) gains are established for certain 'communication with side-information' settings, such as the fading dirty paper channel.