  A self-action problem for a point-like charged particle arbitrarily moving in
flat space-time of six dimensions is considered. A consistent regularization
procedure is proposed which relies on energy-momentum and angular momentum
balance equations. Structure of the angular momentum tensor carried by the
retarded "Li\'enard-Wiechert" field testifies that a point-like source in six
dimensions possesses an internal angular momentum. Its magnitude is
proportional to the square of acceleration. It is the so-called {\em rigid}
relativistic particle; its motion is determinated by the higher-derivative
Lagrangian depending on the curvature of the world line. It is shown that
action functional contains, apart from usual "bare" mass, an additional
renormalization constant which corresponds to the magnitude of "bare" internal
angular momentum of the particle.
