In a Google Slides presentation, is it possible to extract part of an inserted image and use it as a new image?
Google Docs Presentation only has a very limited set of image options ranging only to resizing, rotating, and moving. Any additional image manipulation must be done with other software. Many options are available in this area, from free to big dollars, such as Paint.NET and GIMP up to major programs such as Photoshop. Each of these editors have their own advantages and disadvantages depending on what exactly you need to do. If you only need to do basic cropping then you can use Paint on Windows or Paintbrush on Mac which both come free with your OS.
For a quick and easy solution, you could use a screenshot program such as the Snipping Tool in Windows or Screenshot in Ubuntu/Linux to crop whichever part of the image you'd like to select. Most screenshot programs have the option of selecting just a section of the screen. You'll have to save the cropped image as a new image and manually re-insert it into the Google Docs presentation.