Even after a rain shower, the bottom half of my discs brakes face would show signs of rust..I often wonder how this happens so quickly.
Yes, bare iron or steel will very quickly corrode when exposed to water, hence why a thin layer of rust will form on the brake disc surface when rain or water used for washing your car makes contact with it. This thin layer of rust soon disappears when the brakes are applied and the pads make contact with the brake disc surface.
As for the centre section of the disc that’s attached to the hub and doesn’t get cleaned by the brake pads under braking -
@Betts-4; no lease company should penalise you for this when you return your current car as it’s normal on unprotected iron / steel. If the centre section of the discs on your next car are similarly unprotected and it bothers you, as other have said, remove any surface rust with wet and dry abrasive paper and then paint them with either Hammerite paint or a high temperature paint.