  It is known that no length or time measurements are possible in sub-Planckian
regions of spacetime. The Volovich hypothesis postulates that the
micro-geometry of spacetime may therefore be assumed to be non-archimedean. In
this letter, the consequences of this hypothesis for the structure,
classification, and conformal symmetry of elementary particles, when spacetime
is a flat space over a non-archimedean field such as the $p$-adic numbers, is
explored. Both the Poincar\'e and Galilean groups are treated. The results are
based on a new variant of the Mackey machine for projective unitary
representations of semidirect product groups which are locally compact and
second countable. Conformal spacetime is constructed over $p$-adic fields and
the impossibility of conformal symmetry of massive and eventually massive
particles is proved.
