Sony Mobile has updated the PlayStation Video app to build number 16.0.A.0.2, where the big new feature is a download option allowing you to watch video content offline. The download button will appear on your content and you can choose to download to either internal or external storage. Videos can also be streamed through your mobile data connection or Wi-Fi network. The 11.6MB update is currently rolling to Sony Xperia devices.
Thanks Rimaz!