The download is almost certainly all or nothing.
Several apps will let you download audio files. As long as you can get the link from the site, you can paste it into the URL download feature of whatever audio/vidoe player you like.
I only have a couple of vidoe/audio players. I don't use them, but a quick check shows that VLC has a url download option, but OPlayer only has a play URL feature. GoodReader can also do this, but isn't optimized for listening to audio files. Same for Documents by Readdle.
I use Overcast as my podcast app. It alows downloading podcasts for offline listening.
