The is a preview release of WebDAV Navigator for Android. Some features are missing and only available in the full version.
WebDAV Navigator is a WebDAV client available for iOS and Android and Blackberry.

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The full version includes the following additional features:
Sync :- Full, Contribute and Subscribe
No adverts
Slideshow for remote files
Audio/Voice recording
Built-in Webserver/WebDAV server to share files
Bookmarks for remote and local folders
Unrar, Zip/Unzip support