It only started doing this yesterday, before then it turned off and on at the same time as the ignition!?

it will be directly wired to your battery

Not wired to ignition

Most cars radios do switch off with the ignition - but there is also an option to have them on without the ignition - normally a switch on the back of the unit - something must have triggered this option. You should also be able to switch of the set, usually by pressing the volume button, though.

my girlfriends clio used to do this aswel her ford fiesta does it from time to time aswel l think its a feature that renault have a problem with. she took it to renault and there was a dirty sensor that made it stay on. they cleaned it and the problem went away.

Not an expert, but on my Megane it stays on until drivers door is opened and dont have any reason to think this is a fault, pretty sure thats the way it should be

when you turn ignition off turn the radio off after
it is for convenience more than anything and better for the environment,
it saves you sitting on the drive way with the engine running whilst waiting for the song/news bullitin to finish and also allows you to leave radio on for someone who is not leaving the car but do not trust to leave keys with

My Renault Scénic is radio stays on like this. But turns off as soon as l open the drivers door. It is not a fault on the Scénic and may be the same with the Clio. Try it.

hi my scenic does this & l had a clio that did this aswell its directly connected to the battery its not ignition controled u will probably find if u take out the cd then turn of the ignition it will go off it will only stay on with a cd in the player the same if u turn it off then the next time u get in the car the sterio starts when u put on the ignition even though u turned it off the last time u were in the car if u remove the cd before switching the ignition off the stereo will not start when u turn the ignition on it will come on if a cd is in the player