Yes. What you need are printable globe gores: narrow wedge-shaped slices designed to wrap a flat image onto a sphere.

Based on the answers, there are a few workable options:

• IP-Slicer / sphere-slicer.pl: a script that generates a chosen number of slices from an image. You specify the number of gores and the target sphere circumference.
• USGS Daisy-Petal creator: a tool for creating globe gores, including “half gores” joined at the poles, which can make alignment and gluing easier.
• GIMP with the MathMap extension: can also perform this kind of image transformation.

In practice, use one of these tools to convert the image into gores, print them, cut them out, and glue them carefully onto the sphere. More slices generally fit the sphere more smoothly, but also mean more cutting and alignment work.

If your source is a normal family photo rather than a map, expect distortion near the poles because wrapping a flat image onto a sphere always requires some stretching or splitting.