  We discuss astrophysical implications of $\kappa$-Minkowski space-time, in
which there appears space-time noncommutativity. We first derive a velocity
formula for particles based on the motion of a wave packet. The result is that
a massless particle moves at a constant speed as in the usual Minkowski
space-time, which implies that an arrival time analysis by $\gamma$-rays from
Markarian (Mk) 421 does not exclude space-time noncommutativity. Based on this
observation, we analyze reaction processes in $\kappa$-Minkowski space-time
which are related to the puzzling detections of extremely high-energy cosmic
rays above the Greisen-Zatsepin-Kuzmin cutoff and of high-energy ($\sim$20 TeV)
$\gamma$-rays from Mk 501.
