Can you use split second spells in response to a mana ability?

Mana abilities are special: they do not go on the stack and cannot be responded to by anyone (CR 605.3b). Because they resolve immediately the moment they are activated or triggered, there is simply no opportunity for any player to cast a spell or activate an ability in response — not even a split second spell.

Split second (CR 702.61a) prevents players from casting spells or activating non-mana abilities while a split second spell is on the stack. It has nothing to do with mana abilities. Split second does not grant the ability to respond to things that normally cannot be responded to; it only restricts what others can do while it is on the stack.

To be clear: you cannot use a split second spell to "beat" a mana ability, because mana abilities never give anyone a chance to respond in the first place. The game simply executes the mana ability and moves on.

Concrete example: Your opponent taps a Forest (a mana ability) to add one green mana. You cannot cast Sudden Shock (split second) in response, because the Forest's tap ability resolved instantly — it never went on the stack, so there was never a moment to respond.

The only things that can happen while a mana ability is resolving are other mana abilities (CR 605.3b), and even those are simply executed in sequence rather than placed on a stack.