We study local minimizers of the micromagnetic energy in small ferromagnetic 3d convex particles for which we justify the Stoner-Wohlfarth approximation: given a uniformly convex shape $\Omega \subset {\mathbf{R}}^3$, there exist $\delta_c$>0 and $C > 0$ such that for $0 < \delta \leq \delta_c$ any \textit{local} minimizer $\mathbf{m}$ of the micromagnetic energy in the particle $\delta \Omega$ satisfies $\|\nabla \mathbf{m} \|_{L^2} \leqslant C \delta^2$. In the case of ellipsoidal particles, we strengthen this result by proving that, for $\delta$ small enough, \tmtextit{local} minimizers are exactly spatially uniform. This last result extends W.F. Brown's fundamental theorem for fine 3d ferromagnetic particles [Brown (1968), Di Fratta et al. (2011)] which states the same result but only for \textit{global} minimizers. As a by-product of the method that we use, we establish a new Liouville type result for locally minimizing $p$-harmonic maps with values into a closed subset of a Hilbert space. Namely, we establish that in a smooth uniformly convex domain of $\mathbf{R}^d$ any local minimizer of the $p$-Dirichlet energy ($p > 1$, $p \neq d$) is constant.