The center will remain in place but this is not a problem.

Press the > arrow key (a few times) and you should see the ball fall. Click 'enable softbody' and then turn up 'Grav' to 9.8. Now for the soft body select the ball and go to the object tab then the physics subtab or whatever your version has. Gravity acts on the z-axis (sphere should be above the plane relative to the z axis). Make sure to do this in the right views so that it is aligned properly. Move the sphere up and place a plane below it. A cube sphere is made by subdividing a cube and doing a 'To Sphere' in the Edit window, under Mesh Tools. I would use a cube sphere or an icosphere, UV spheres don't deform well as they have too few vertices. We are going to make a big rubber ball, but not a big bouncy one, a flat (and somewhat lifeless) one. The cotton one would only stretch a little bit, so the vertices would stay essentially the same distance apart. The elastic one has edges that can extend, so if you view them in wire-frame (with vertices and edges visible) you would see the edges are more extended than an equal distance. To put this in perspective, picture two cloths, one elastic and one cotton. For example, you can make it so edges can stretch really far (aka elasticity), allowing the vertices to become distant, or you can make the edge stiff, so the vertices will always stay the same distance apart. By adjusting the settings, you change the behaviour of the edges that connect the vertices. Softbody makes each individual vertex its own object that falls according to gravity and reacts to outside forces like fields. If the menus don't agree then just do what seems logical. Due to rapid updates in Blender, some menu items may be different from those mentioned here.

Blender physics drop object