ABSTRACT Observations of molecular gas near the Galactic Centre (|l| < 10°, |b| < 1°) reveal the presence of a distinct population of enigmatic compact clouds that are characterized by extreme velocity dispersions ($Δ v \gt 100\, {\rm km\, s^{-1}}$). These extended velocity features are very prominent in the data cubes and dominate the kinematics of molecular gas just outside the Central Molecular Zone (CMZ). The prototypical example of such a cloud is Bania Clump 2. We show that similar features are naturally produced in simulations of gas flow in a realistic barred potential. We analyse the structure of the features obtained in the simulations and use this to interpret the observations. We find that the features arise from collisions between material that has been infalling rapidly along the dust lanes of the Milky Way bar and material that belongs to one of the following two categories: (i) material that has ‘overshot’ after falling down the dust lanes on the opposite side; (ii) material which is part of the CMZ. Both types of collisions involve gas with large differences in the line-of-sight velocities, which is what produces the observed extreme velocity dispersions. Examples of both categories can be identified in the observations. If our interpretation is correct, we are directly witnessing (a) collisions of clouds with relative speeds of $∼ 200\, {\rm km\, s^{-1}}$ and (b) the process of accretion of fresh gas onto the CMZ.