We study push-pull rumour spreading in ultra-small-world models for social networks where the degrees follow a power-law distribution. In a non-geometric setting, Fountoulakis, Panagiotou and Sauerwald have shown that rumours always spread ultra-fast (SODA 2012). On the other hand, Janssen and Mehrabian have found that rumours spread slowly in a spatial preferential attachment model (SIDMA 2017). We study the question systematically for the model of Geometric Inhomogeneous Random Graphs (GIRGs), which has been found to be a good theoretical and empirical fit for social networks. Our results are two-fold: first, with classical Euclidean geometry slow, fast and ultra-fast (i.e., polynomial, polylogarithmic and doubly logarithmic number of rounds) rumour spreading may occur, depending on the exponent of the power law and the strength of the geometry in the network, and we fully characterise the phase boundaries between these regimes. The regimes do not coincide with the graph distance regimes, i.e., polylogarithmic or even polynomial rumour spreading may occur even if graph distances are doubly logarithmic. We expect these results to hold with little effort for related models, e.g. Scale-Free Percolation. Second, we show that rumour spreading is always (at least) fast in a nonmetric geometry. The considered non-metric geometry allows to model social connections where resemblance of vertices in a single attribute, such as familial kinship, already strongly indicates the presence of an edge. Classical Euclidean geometry fails to capture such ties.