  The HST treasury program 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 6arcmin x 6arcmin
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 $\sim 5 \times 10^{13}M_{\odot}$. Combining the
total mass distribution derived from lensing with multi-wavelength data, we
evaluate the physical significance of each candidate substructure, and conclude
that 5 out of the 7 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.
