In this paper we prove the existence of at least three solutionsto the following Kirchhoff nonlocal fractional equation:\begin{equation*} \begin{cases} M \left (\int_{\mathbb{R}^n\times \mathbb{R}^n} |u (x) - u (y)|^2 K (x - y) d x d y - \int_\Omega |u (x)|^2 d x \right) ((- \Delta)^s u - \lambda u) \\ \hspace{2cm} \in \theta (\partial j (x, u (x)) + \mu \partial k (x, u (x))), & \textrm{in}\;\; \Omega,\\ u = 0, & \textrm{in}\;\; \mathbb{R}^n \setminus \Omega, \end{cases}\end{equation*}where $(- \Delta)^s$ is the fractional Laplace operator, $s \in(0, 1)$ is a fix, $\lambda, \theta, \mu$ are real parameters and$\Omega$ an open bounded subset of $\mathbb{R}^n$, $n > 2 s$, withLipschitz boundary. The approach is fully based on a recent threecritical points theorem of Teng [K. Teng, Two nontrivial solutionsfor hemivariational inequalities driven by nonlocal ellipticoperators, Nonlinear Anal. (RWA) 14 (2013) 867-874].