  In the classical vacuum Maxwell-Lorentz theory the self-force of a charged
point particle is infinite. This makes classical mass renormalization necessary
and, in the special relativistic domain, leads to the Abraham-Lorentz-Dirac
equation of motion possessing unphysical run-away and pre-acceleration
solutions. In this paper we investigate whether the higher-order modification
of classical vacuum electrodynamics suggested by Bopp, Lande, Thomas and
Podolsky in the 1940s, can provide a solution to this problem. Since the theory
is linear, Green-function techniques enable one to write the field of a charged
point particle on Minkowski spacetime as an integral over the particle's
history. By introducing the notion of timelike worldlines that are "bounded
away from the backward light-cone" we are able to prescribe criteria for the
convergence of such integrals. We also exhibit a timelike worldline yielding
singular fields on a lightlike hyperplane in spacetime. In this case the field
is mildly singular at the event where the particle crosses the hyperplane. Even
in the case when the Bopp-Podolsky field is bounded, it exhibits a directional
discontinuity as one approaches the point particle. We describe a procedure for
assigning a value to the field on the particle worldline which enables one to
define a finite Lorentz self-force. This is explicitly derived leading to an
integro-differential equation for the motion of the particle in an external
electromagnetic field. We conclude that any worldline solutions to this
equation belonging to the categories discussed in the paper have continuous
4-velocities.
