Are there semiconductor materials with hole mobility greater than electron mobility?
The question is complicated because on a very real level a hole is a "real" particle, even though traditional models consider it an absence of electron and this can be used to explain many electrical properties. It has its own wavefunction and therefore physically exists (for lack of better term) due to particle-wave duality. It interacts differently and beyond my intuition to consider what properties a material must have to make hole mobility larger. 

I don't want to engage any further in the physics, it is beyond my expertise and training.  

However, as possible exotic counterexample. Graphene can be created and used in a way that has semiconductor properties, it is hypothetically possible to tune the band gap to any value, some citations suggest that in this exotic material hole and electron mobility is the same. This knowledge is from academic seminars, but not referenced.  The only citation I was able to find was from wikipedia  sourced to this book https://books.google.com/books?id=ammoVEI-H2gC