Water dripping from basement ceiling during heavy rain
I know you say the guttering is fine, but my advice to all homeowners is: "never check the gutters when they are dry". Always check them when it's pouring with rain, the harder the better. There is no better way of seeing exactly where the water is going. 

My thoughts are that if the roof or chimney were leaking, you'd see evidence long before the water hits the basement ceiling, no?

The fact the water leak is above the ground level means the water is likely getting past the cladding. The fact it only does it during extreme rain points to the gutter system overflowing (IMHO). 

The problem may then be twofold; the gutter (as it is) can't cope with heavy rain, and second, there is a breach in the cladding and/ or membrane behind the cladding.