How unusual, not something I've experieced before.

I would treat that with oodles of dash shine on a daily basis for a few days using a product such as Armor All Protectant.

You should not clean car plastics with detergents, try feeding the the area with some proper polish meant for car interiors, i find Autosmart's Finish excellent for this but there are many products available..

On external plastics which have gone like this i have read of people using Linseed oil to 'feed' the plastic with good results, but not sure if the same would apply to interior plastics.

I have used peanut butter as advised on a detailing forum and following that to reduce the mess, peanut oil. You can laugh, but it works!

I've heard about using peanut butter but never tried it, I have used heat from a hair dryer though to bring oils back to the surface on plastics, I've also used autofiness "revive " not cheap but it's brilliant.