  Discrete time quantum walks are unitary maps defined on the Hilbert space of
coupled two-level systems. We study the dynamics of excitations in a nonlinear
discrete time quantum walk, whose fine-tuned linear counterpart has a flat band
structure. The linear counterpart is, therefore, lacking transport, with exact
solutions being compactly localized. A solitary entity of the nonlinear walk
moving at velocity $v$ would therefore not suffer from resonances with small
amplitude plane waves with identical phase velocity, due to the absence of the
latter. That solitary excitation would also have to be localized stronger than
exponential, due to the absence of a linear dispersion. We report on the
existence of a set of stationary and moving breathers with almost compact
superexponential spatial tails. At the limit of the largest velocity $v=1$ the
moving breather turns into a completely compact bullet.
