Approaching space-time through velocity in doubly special relativity

We discuss the definition of velocity as $dE/d|\mathbf{p}|$, where $E$, $\mathbf{p}$ are the energy and momentum of a particle, in doubly special relativity (DSR). If this definition matches $dx/dt$ appropriate for the space-time sector, then space-time can in principle be built consistently with the existence of an invariant length scale. We show that, within different possible velocity definitions, a space-time compatible with momentum-space DSR principles cannot be derived.