TD5 Defender immobiliser rapidly flashing when button pressed?

As it says on the tin really. The immobiliser light has a slow repetitive flash when locked, open up a door and the light comes on and stays on, usually pressing the fob button turns it off.
But now: when I press the button, the immobiliser light blinks rapidly, the engine starts, but the light still goes for about 2-3 seconds after the engine starts.

Is this an error code of some sort, or just a warning that the fob battery is low? (I don't have any to hand to try it with, and nor do I have spare fob)

Flat key fob battery, possibly?

Ok double checked this morning. I put the key in the ignition and press the fob button (pressing the button does nothing otherwise - a new development) and the red light goes "flash-flash" 19 times (it might be 20, will count again later)
Any ideas?

@Dobbs, thanks for the suggestion. Will find out what battery it needs and change it. Will I need to pair the key with the car again after that?!

Sorry was wrong there. its not that pressing the button does nothing, but that I have to press it twice before it does anything, as if it's intentionally like that rather than not registering the first press.

Does anyone know what battery these take?