SetOptions[Rasterize, {RasterSize -> 512, ImageSize -> 512}];
SetOptions[Graphics, ImageSize -> 512];
SetOptions[Graphics3D, {ImageSize -> 512, Boxed -> False}];

AllCountries = Union[Take[CountryData["Countries"], All], {"Antarctica"}]

Rasterize[Graphics[{
EdgeForm[Black],
{Hue[Random[]],CountryData[#, "SchematicPolygon"]}& /@ 
AllCountries
}]]

sphere={RGBColor[0.75,0.75,1],EdgeForm[],
{m,n}={32,48};
s=Table[
z=N[-1+2i/m]; r=Sqrt[1-z^2]; t=N[2Pi j/n];
Point @@ (0.99{r Cos[t], r Sin[t], z}),
{i, 0, m}, {j, 0, n}
];
Table[
Polygon[{s[[i+1,j+1]], s[[i+1,j]], s[[i,j]], s[[i,j+1]]}],
{i,m}, {j,n}
]};

map = {
EdgeForm[Black],
{
Hue[Random[]],
Apply[
Function[
{lon, lat},
Point[Cos[lat Degree]Cos[lon Degree], Cos[lat Degree]Sin[lon Degree], Sin[lat Degree]]
],
CountryData[#, "SchematicPolygon"],
{3}
]
}& /@ AllCountries
};

ImageCrop[Rasterize[Graphics3D[{
RGBColor[0.75,0.75,1], Sphere[{0,0,0}, 0.99],
map /. Point -> List
}]]]

ImageCrop[Rasterize[Graphics3D[{
RGBColor[0.75,0.75,1], Cuboid @@ (0.99{{-1,-1,-1}, {1,1,1}}),
map /. Point[x_,y_,z_] :> {x,y,z}/Max[Abs[{x,y,z}]]*(x^2+y^2+z^2)
}]]]