Our service consumers often ask the same question. Direct answer: No. Indirect answer yes. A few options:
(1) Consume the image service in an ArcGIS Desktop client (ArcMap or ArcPro). From within the client, the user can access client based tools to download rasters.
(2) Access the service URL and construct a query to download rasters following the instructions for the Esri REST API.
(3) Create an application to allow raster tile downloads. There are a few web app builder templates that may work.
(4) Workaround this limitation with a hybrid solution of your own making. Here's what we did:
Maine Orthoimagery Regional 2017 (Imagery Layer) NOTE: The Orthoimagery Download and Discovery App under the Tile Download Link URL in our description leads to an Esri Mature Support code base Web App that is really unsupported. We're looking into options to replace it but haven't a clear direction yet. Esri Excalibur is under consideration.