We study the fractional p-Kirchhoff equation $$ \Big( a+b \int_{\mathbb{R}^N}{\int_{\mathbb{R}^N}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\, dx\, dy\Big) (-\Delta)_p^s u-\mu|u|^{p-2}u=|u|^{q-2}u, \quad x\in\mathbb{R}^N, $$ where \((-\Delta)_p^s\) is the fractional p-Laplacian operator, a and b are strictly positive real numbers, \(s \in (0,1)\), \(1 < p< N/s,\) and \(p< q< p^*_s-2\) with \(p^*_s=\frac{Np}{N-ps}\). By using the variational method, we prove the existence and uniqueness of global minimum or mountain pass type critical points on the \(L^p\)-normalized manifold\(S(c):=\big\{u\in W^{s,p}(\mathbb{R}^N): \int_{\mathbb{R}^N} |u|^pdx=c^p\big\}\).