The HST treasury programme BUFFALO provides extended wide-field imaging of the six Hubble Frontier Fields galaxy clusters. Here we present the combined strong and weak-lensing analysis of Abell 370, a massive cluster at z = 0.375. From the reconstructed total projected mass distribution in the 6 arcmin x 6 arcmin BUFFALO field-of-view, we obtain the distribution of massive substructures outside the cluster core and report the presence of a total of seven candidates, each with mass & SIM;5 x 10(13) M-& ODOT;. Combining the total mass distribution derived from lensing with multiwavelength data, we evaluate the physical significance of each candidate substructure, and conclude that five out of the seven substructure candidates seem reliable, and that the mass distribution in Abell 370 is extended along the north-west and south-east directions. While this finding is in general agreement with previous studies, our detailed spatial reconstruction provides new insights into the complex mass distribution at large cluster-centric radius. We explore the impact of the extended mass reconstruction on the model of the cluster core and in particular, we attempt to physically explain the presence of an important external shear component, necessary to obtain a low root-mean-square separation between the model-predicted and observed positions of the multiple images in the cluster core. The substructures can only account for up to half the amplitude of the external shear, suggesting that more effort is needed to fully replace it by more physically motivated mass components. We provide public access to all the lensing data used as well as the different lens models.