Infinitesimal thickening of a smooth closed subscheme

Question

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension of algebras $$ 0 \to I/I^2 \to A/I^2 \to A/I \to 0 $$ splits (as a sequence of algebras)? Do I need $A$ to be smooth or $I/I^2$ to be projective? In other words can we say that infinitesimal thickening of a smooth subscheme is just adding direction in the conormal bundle?