I noticed an update for the BBC iPlayer android app this morning and noticed something quite significant in the 'what's new' section.
You can now download videos to watch later or offline. This is a fantastic update, especially with holidays coming up.
I'm not sure if it will keep the shows for 7 days or the full 30 (like the new iPlayer online http://www.bbc.co.uk/iplayer), but will give it a try.
The downside is this is only for android devices from 4.0 and up. That means it won't be available for the, roughly, 15-20% of devices on older android OS.
The app can be downloaded from Google Play here.