We are concerned with the existence and local uniqueness of normalized k-peak solutions to the following fractional nonlinear Schrödinger equation \(\begin{aligned} (-\Delta )^s u+V(x)u= au^{p}+\mu u,\ x\in \mathbb {R}^N \end{aligned}\) with constraint \(\int _{\mathbb {R}^N}u^2(x)dx=1\) , where V(x) is a degenerated trapping potential with non-isolated critical points. The main feature of this work is to fully exploit the nonlocal properties of the fractional operator \((-\Delta )^s\) , which makes all the difference in overcoming all difficulties coming from the algebraic decay at infinity of the ground states of the limiting problem, the indispensable point-wise estimates of the solutions when establishing the local Pohozaev identities, and the influence of the mass constraint in performing reduction and blowing up analysis. We consider the nonlocal problem both in the direct form and in its equivalent harmonic extension case, optimizing the range of s and distinguishing all the cases of \(p-1<\frac{4s}{N}\) , \(p-1=\frac{4s}{N}\) , and \(p-1>\frac{4s}{N}\) , which are called respectively the mass-subcritical, the mass-critical, and the mass-supercritical case.
