  Quantum spin models have been studied extensively in one and higher
dimensions. Furthermore, these systems have been doped with holes to study
$t$--$J$ models of $SU(2)$ spin-1/2. Their anyonic counterparts can be built
from non-Abelian anyons, such as Fibonacci anyons described by $SU(2)_3$
theories, which are quantum deformations of the $SU(2)$ algebra. Inspired by
the physics of $SU(2)$ spins, several works have explored ladders of Fibonacci
anyons and also one-dimensional (1D) $t$--$J$ models. Here we aim to explore
the combined effects of extended dimensionality and doping by studying ladders
composed of coupled chains of interacting itinerant Fibonacci anyons. We show
analytically that in the limit of strong rung couplings these models can be
mapped onto effective 1D models. These effective models can either be gapped
models of hole pairs, or gapless models described by $t$--$J$ (or modified
$t$--$J$--$V$) chains of Fibonacci anyons, whose spectrum exhibits a
fractionalization into charge and anyon degrees of freedom. By using exact
diagonalizations for two-leg and three-leg ladders, we show that indeed the
doped ladders show exactly the same behavior as that of $t$--$J$ chains. In the
strong ferromagnetic rung limit, we can obtain a new model that hosts two
different kinds of Fibonacci particles - which we denote as the heavy $\tau$'s
and light $\tau$'s. These two particle types carry the same (non-Abelian)
topological charge but different (Abelian) electric charges. Once again, we map
the two-dimensional ladder onto an effective chain carrying these heavy and
light $\tau$'s. We perform a finite size scaling analysis to show the
appearance of gapless modes for certain anyon densities whereas a topological
gapped phase is suggested for another density regime.
