– make the sphere editable
– select all the sphere’s polys
– select Disconnect BUT do it by the gear that’s beside the word ‘disconnect’ -that gets to the settings for that command
– for Disconnect there is only one setting, ‘Preserve Groups’ uncheck that
– now do a Normal Scale
– now Extrude – right-click or “D” key (with or without Create Caps checked, depending on what you need.