Start with a hemisphere with a picture on it. Partition it into regions, then make pyramids (with bases with spherical caps) by extruding (intruding) the regions to the center. If the pieces are designed to fit together in only one way, then it can be a jigsaw puzzle, even having shapes that stay together if correctly matched (e.g., dovetail).
Two hemispherical puzzles (the pieces of both jumbled together) could, after both being solved, be joined together into a sphere.
All the pieces could be shaped the same, but this is much harder for a hemisphere than for a sphere. Maybe a special base, though that will give away most of the challenge of figuring out how the pieces are supposed to fit.
Trim off or round off the tips of the pyramids to make the pieces less dangerous to accidentally step on.