You cannot silence the main alarm function from the key fob as this deletes the point of the alarm(thatcham???), BUT my dad had this some 2 months or so ago on his 4i, and it turned out to be a very small amount of corrosion on the battery terminal between the round post and the clamp and no I could not beleive it either.
It worked perfect, armed,disarmed, triggered etc but no sound, battery ok car starting perfect, need a hefty starter and battery for 4 litre anyway but no sound, thought it worth a go so cleaned up terminal and perfection right away.
tabetha
ps ai606 as well