Water – Are the basement walls sweating

Yes, drylok can be recoated easily. Depending how bad (or not-bad) the spots are, you might think about whether the super-thick "masonry waterproofer" or the "concrete floor paint" is more appropriate. If staining appears to indicate leaks, probably the thick stuff.

Ease of re-coat is one of several reasons the concrete floor paint is on my shop floor rather than some fancy epoxy product. 5-6X cost differential and not being slippery (without having to add some surface treatment) are two of the others that spring to mind. And, of course, not being gray concrete...

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.