Lads lads lads.
Thanks for getting involved, but boy this got heated! Thank you all for the input.
As well as the suggestions here, I've had a suggestion of a stub wall up against my wall with a DPM vertically between it and my wall for the length of the wall. The outside of the new wall would be on the boundary, and damp couldn't get through it to my extension wall. This sounds good. This is similar to the concrete gravel board suggestion (thanks @Gone West) (again with DPM vertically between it and my wall could work).
@ETC thanks for the diagram, that captures it.
Something physical is probably the only realistic solution, because even if he digs it out now, over time it's going to fill back up with crud. Maybe not today, maybe not tomorrow, but some day in the future, I'll have to crack this nut again. Damp waits for no man and soil creeps. (There's also the added benefit of demarcating the boundary.)
@saveasteading I'm not sure I fully understood your suggestion of leveling off both mine and his land, if he doesn't want his level brought down?