Hi there,
I've got exactly the same problem on a Baxi Solo 2 PF40.
Found a number of threads in the forum on DIYnot, and quite a few suggested that the PCB had failed.
I've been lucky and have located the fault to the relay that supplies the pump. Here's the symptoms I found:
BE CAREFUL DURING YOUR INVESTIGATIONS BECAUSE THERE'S 240V AC ON THE CIRCUIT BOARD - ONLY INVESTIGATE IF YOU KNOW WHAT YOU ARE DOING:
Symptoms: Pump stays on 24 hrs with central heating and hot water both turned off.
Opened up the front of the boiler, and confirmed that 240V AC supply to pump was live.
Removed power, and gained access to PCB. Located relay that serves the pump on the PCB (black cube nearest supply to pump). Found that power to the pump is Normally-On. This means that the PCB has to energise the relay to turn the power to the pump off.
Turned power back on, and VERY CAREFULLY checked the voltage on the coil side of the relay. Found 10.6V DC. This meant that the PCB was trying to turn the pump off, but couldn't because the relay must be sticking.
So, I'm about to look for a new relay! Feeling confident that it'll fix it, hence why I'm posting my (potentially premature) message up here.
Good luck!