It is proposed how the Extended Relativity Theory in $C$-spaces (Clifford spaces) allows a unified formulation of point particles, strings, membranes and $p$-branes, moving in ordinary target spacetime backgrounds, within the description of a single $polyparticle$ moving in $C$-spaces. The degrees of freedom of the latter are provided by Clifford polyvector-valued coordinates (antisymmetric tensorial coordinates). A correspondence between the $p$-brane ($p$-loop) wave functional ``Schroedinger-like" equations of Ansoldi-Aurilia-Spallucci and the polyparticle wave equation in $C$-spaces is found via the polyparticle/$p$-brane duality/correspondence. The crux of exploiting this correspondence is that it might provide another unexplored avenue to quantize $p$-branes (a notoriously difficult and unsolved problem) from the more straightforward quantization of the polyparticle in $C$-spaces, even in the presence of external interactions. We conclude with some comments about the $compositeness$ nature of the polyvector-valued coordinate operators in terms of ordinary $p$-brane coordinates via the evaluation of $n$-ary commutators.