Context. We present the study of a magnitude limited sample (m_B ≤ 16.6 mag) of 13 late type galaxies (LTGs), observed inside the virial radius, R_vir ∼ 0.7 Mpc, of the Fornax cluster within the Fornax Deep Survey (FDS). Aims. The main objective is to use surface brightness profiles and g − i colour maps to obtain information on the internal structure of these galaxies and find signatures of the mechanisms that drive their evolution in high-density environments inside the virial radius of the cluster. Methods. By modelling galaxy isophotes, we extract the azimuthally averaged surface brightness profiles in four optical bands. We also derive g − i colour profiles, and relevant structural parameters like total magnitude and effective radius. For ten of the galaxies in this sample, we observe a clear discontinuity in their typical exponential surface brightness profiles, derive their “break radius”, and classify their disc-breaks into Type II (down-bending) or Type III (up-bending). Results. We find that Type II galaxies have bluer average (g − i) colour in their outer discs while Type III galaxies are redder. The break radius increases with stellar mass and molecular gas mass while it decreases with molecular gas-fractions. The inner and outer scale-lengths increase monotonically with absolute magnitude, as found in other works. For galaxies with CO(1-0) measurements, there is no detected cold gas beyond the break radius (within the uncertainties). In the context of morphological segregation of LTGs in clusters, we also find that, in Fornax, galaxies with morphological type 5 < T ≤ 9 (∼60% of the sample) are located beyond the high-density, ETG-dominated regions, however there is no correlation between T and the disc-break type. We do not find any correlation between the average (g − i) colours and cluster-centric distance, but the colour-magnitude relation holds true. Conclusions. The main results of this work suggest that the disc-breaks of LTGs inside the virial radius of the Fornax cluster seem to have arisen through a variety of mechanisms (e.g. ram-pressure stripping, tidal disruption), which is evident in their outer-disc colours and the absence of molecular gas beyond their break radius in some cases. This can result in a variety of stellar populations inside and outside the break radii.