Take a regular icosahedron. For each face, take the image of the circumcenter of the icosahedron by the reflection about the plane containing the face. Now subtract the sphere centered at this point with radius the circumradius to the icosahedron. Here is the result:
Is it a true hyperbolic icosahedron?
There's a second question. This construction does not work for a regular tetrahedron (the result is far from being a hyperbolic tetrahedron). Why?