  We use the Cluster-EAGLE simulations to explore the velocity bias introduced
when using galaxies, rather than dark matter particles, to estimate the
velocity dispersion of a galaxy cluster, a property known to be tightly
correlated with cluster mass. The simulations consist of 30 clusters spanning a
mass range $14.0 \le \log_{10}(M_{\rm 200c}/\mathrm{M_\odot}) \le 15.4$, with
their sophisticated sub-grid physics modelling and high numerical resolution
(sub-kpc gravitational softening) making them ideal for this purpose. We find
that selecting galaxies by their total mass results in a velocity dispersion
that is 5-10 per cent higher than the dark matter particles. However, selecting
galaxies by their stellar mass results in an almost unbiased ($<5$ per cent)
estimator of the velocity dispersion. This result holds out to $z=1.5$ and is
relatively insensitive to the choice of cluster aperture, varying by less than
5 per cent between $r_{\rm 500c}$ and $r_{\rm 200m}$. We show that the velocity
bias is a function of the time spent by a galaxy inside the cluster
environment. Selecting galaxies by their total mass results in a larger bias
because a larger fraction of objects have only recently entered the cluster and
these have a velocity bias above unity. Galaxies that entered more than $4 \,
\mathrm{Gyr}$ ago become progressively colder with time, as expected from
dynamical friction. We conclude that velocity bias should not be a major issue
when estimating cluster masses from kinematic methods.
