You can use the Buffer tool to add buffers to create a new shapefile representing the 1 mile buffers.
Then you can use the Extract by Mask tool to clip the raster inside of the buffer shapefile features. Mind you need a Spatial Analyst extension to use this tool.
I recommend using Python or even ModelBuilder to create a loop that will step through each feature in your buffer shapefile, and clip your raster, resulting in individual rasters for each feature that was buffered. I post another link, but I do not have a high enough reputation. Simply Google "ArcGIS 10.1 ModelBuilder Iterators".