In this work, we introduce a stochastic growth-fragmentation model for the expansion of the network of filaments, or mycelium, of a filamentous fungus. In this model, each individual is described by a discrete type e ∈ {0, 1} indicating whether the individual corresponds to an internal or terminal segment of filament, and a continuous trait x ≥ 0 corresponding to the length of this segment. The length of internal segments cannot grow, while the length of terminal segments increases at a deterministic speed v. Both types of individuals/segment branch according to a type-dependent mechanism. After constructing the stochastic bi-type growth-fragmentation process of interest, we analyse the corresponding mean measure (or first moment semigroup) and show a Harris-type ergodic theorem stating that, in the long run, the total mass of the mean measure increases expontially fast while the type-dependent density in trait stabilises to an explicit distribution. In the particular model we consider, which depends on only 3 parameters, all the quantities needed to describe this asymptotic behaviour are explicit, which paves the way for parameter inference based on data collected in lab experiments.