Fisher and Paykel Smart Drive Pride 5.5 washing machine will not pump water out...makes a beeping noise when trying to pump water out.

That sounds indicative or either a failed controller board or bad connections on it. What you can try if you wish is getting to the control board and spraying all the connecters with electrical switch cleaner. For any board edge connectors remove the connector, spray the contacts on the board edge and polish the board edge contact strips with a piece of kitchen towel. If that doesn't fix it call for service and diagnosis as control boards are expensive, so you need not only a correct diagnoses but warranty on the part and the labour.

The other possibility as an afterthought, as the drain blockage caused this, is the pump working okay? These machines have self test procedures on the pump, and if the pump doesn't pass, the machine shuts down. If the drain was blocked and the pump was running for an extending time, it may be heat stressed. That would be a much cheaper repair.

Further ideas in this link

https://www.google.com/search?q=Fisher+%26+Paykel+Smart+Drive+Pride+5.5+Top+Load+turns+off+after+filling

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Either the cold water inlet solenoid has failed or there is a fault with the electronic control module, Unfortunately the only way to find out is to replace the cold water solenoid and see if it works, if it doesn't then the electronic control board is faulty.

does it flood if left running ?
if so replace the cold water solenoid

this sounds like the motor controller cant start the motor. I would be looking at the motor controller itself.
when it's filling with water, does the bowl go around slowly?

The magnetic reed lid switch could be faulty.

The pump is over heating and cutting power.

most likely you have a bad drain pump.or its possible you have foreign objects inside the drain pump preventing it from operating

blocked ,jammed or fulty drain pump

Probably a bad controller. Note the lights that are illuminated and call F/P.