Citroen c5 engine cooling fan staying on even with engine switched off

When the Automatic Climate Control is set, you are no longer in control of settings and fan speeds. The sensors make the unit run at full speed and don't power down until the set temperature is reached. If you have the temperature set to 18'C (Max Cold), it is likely to never achieve that. If you want to take control back, you need to take it from Auto, to Manual. If it still does it, you need to go to an auto electrician who will use a pro grade scan tool to check on communications to the Body Control Module and the sensors controlling AC, Heating, and Fan Speeds.

This is the Owner Manual covering Climate Control

https://www.manualslib.com/manual/397817/Citroen-C5.html?page=105

This is a technical manual on the D-Series explaining AC, I just found it interesting.

https://www.manualslib.com/manual/1061220/Citroen-Ds-Series.html?page=50#manual

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There is a temp sensor threaded into the radiator, it will need to be replaced. To make sure this is needed, you can locate it and unplug it. It will have two wires in this plug. Short the two together, be careful because the fan may turn on even if the key is off. Some run till cool after engine has been turned off.

take it in for a charging system check as it indicates that the alternator is not providing enough current to run the system
possible voltage regulator problem and if that unit is part of the ECM then it will be a replacement ECM

replace the fan motor and check for burnt out speed control resistors foe the fan --in the same area as the fan and motor

depending on the engine temperature the fan is supposed to remain on after you turn the engine off, until the engine reaches a preset cooler temperature. when it reaches cooler temperature usually within 5 minutes or so it should cut off... if you cut it off and it is running 30 minutes later.. then there is a problem with the fan relay or temperature sending unit.

fans are designed to run on after you shut engine down as engine temperature rise after switching off . running hot could be caused by the top hose having gone soft when you rev the engine the water pump sucks the hose flat stopping the coolant flow or the radiator is blocked and need flushing out

Presuming your car is fitted with the automatic heating/cooling system.
There is only one fan controls the heating/cooling system the distribution from this to the various vents is controlled by a series of electronic flaps which open or close dependent on the temperature selected at the control unit.
If the motor is running and the drivers side is receiving cooling it may be a problem with the control unit or a flap has stuck open.
Replace the control unit first as the flaps are a citroen job.
Control unit is the bit with the temperature displays etc.

I am not familiar with the X5, but most cars electric fans are controlled by a thermostat. If the thermostat, or the associated relay is stuck or defective, this could cause this problem. BE AWARE however that most fans stay on for a time to cool the engine, even after shut down. If your battery is weak, the fan staying on could draw the battery down.

Not a Citroen expert here, but thats usually a stuck cooling fan relay.