We are using a map that has a tiled image service as the basemap. It's on our own server, published with 10.2.2 with the option enabled to allow exporting. On the device, the download icon is available, and we can click on it to select the area to download. However the success stops there. Usually you get the little toggle at the bottom (Work Area/Map Detail) but that is not present. You can click on the "Download" link at the top right but nothing happens. This has worked fine with our tiled map services, so I wonder if the fact that it's the image service is the problem? I cannot find anything in the documentation that addresses this.
You should be able to download your own tiled layers, they don't even have to be the basemap layer.
The WorkArea/MapDetail control at the bottom doesn't appear until the map finishes loading. All the layers have to finish loading before that will appear. The Download button won't do anything until after both Work Area and Map Detail have been set.
Can you tell if all the layers had finished loading?
I have given it quite a bit of time and not seeing any different results. The map only has 1 feature layer (with only a few features thus far) and the imagery. Can I invite you to my group and see if it works for you?
The problem is the basemap. It is not a tiled layer, it is an image service layer which cannot be taken offline. Only tiled layers can be taken offline currently.
If you can create a tile package on your desktop then that can be side-loaded into Collector, but Collector cannot create tile packages from image services.