  We introduce a stabilizer code model with a qutrit at every edge on a square
lattice and with non-invertible plaquette operators. The degeneracy of the
ground state is topological as in the toric code, and it also has the usual
deconfined excitations consisting of pairs of electric and magnetic charges.
However, there are novel types of confined fractonic excitations composed of a
cluster of adjacent faces with vanishing flux. They manifest confinement, and
even larger configurations of these fractons are fully immobile although they
acquire emergent internal degrees of freedom. Deconfined excitations change
their nature in presence of these fractonic defects. As for instance, fractonic
defects can absorb magnetic charges making magnetic monopoles exist while
electric charges acquire restricted mobility. Furthermore, some generalized
symmetries can annihilate any ground state and also the full sector of fully
mobile excitations. All these properties can be captured via a novel type of
\textit{non-commutative} and \textit{non-Abelian} fusion category in which the
product is associative but does not commute, and can be expressed as a sum of
(operator) equivalence classes. Generalized non-invertible symmetries give rise
to the feature that the fusion products form a non-unital category without a
proper identity. We show that a variant of this model features a deconfined
fracton liquid phase and a phase where the dual (magnetic) strings have
condensed.
