The Poisson gauge theory is a semi-classical limit of full non-commutative gauge theory. In this work we construct an L f ull ∞ algebra which governs both the action of gauge symmetries and the dynamics of the Poisson gauge theory, including the gauge covariant objects like the Poisson field strength and the gauge covariant derivative of matter field. We derive the minimal set of non-vanishing ℓ-brackets and prove that they satisfy the corresponding homotopy relations. On the one hand, it provides new explicit non-trivial examples of L ∞ algebras. On the other hand, it can be useful for bootstrapping the full non-commutative gauge theory. In addition we show that the derivation properties of ℓ-brackets on L f ull ∞ with respect to the truncated product on the exterior algebra are satisfied only for the canonical non-commutativity. In general, L f ull ∞ does not have a structure of P ∞ algebra.