Just to save everyone some typing, I know a GFCI works and I've installed dozens of them over time, both as standalone devices and as protection for downstream receptacles. Line vs. load, Pre-2006 devices, Post-2006 devices, etc.
Anybody ever heard of a case where the GFCI acts like this:
Press the test button and the GFCI "trips" - the button pops out and the downstream receptacles go dead. However, the GFCI itself is still live.
A tester shows that the GFCI is wired correctly, at least as far as hot, neutral and ground are concerned. I can only assume that something is wrong internal to the GFCI but if that's the case, that's a really dangerous failure mode.
The homeowner doesn't know when the device was installed and isn't a family member or close enough friend for me to start messing with wiring in a 1940's era house. I shut off the breaker and told her to call an electrician.
I've seen GFCI's fail, but not like that.