No, this is not possible in WebDrive.

However, you can mark certain folders/files available for offline use (right-click --> WebDrive --> Make Available Offline). This will download the folder(s) and file(s) into cache and they will not be flushed from cache in normal operation. In order to remove such folders/files from the cache, you can either remove them from offline use, or you can go to the Cache folder in Explorer and manually delete the cache.