duh, the sun fight requires you to use the jump button to shoot
and, you can't get 32 ice creams without pressing space at least once.
yes, i found that out.
You only have to press A once to do a ledge climb when you have the flamethrower but you have to press A a bunch of times to beat the sun I don't think the sun or the ledge climb should count because they aren't really jumps and another reason for the ledge climb: you can do it with the X button without any powerups, but since pressing X activates the powerup ability, the only thing that works is pressing A. So, you're doing an action that doesn't have to be done with the A button, but you're doing it with the A button because you can't use another button (the X button)
if you mean ledge climb by holding onto a ledge and jumping to get up you could always press shift and spin :|