In this paper, thick branes generated by the mimetic scalar field with Lagrange multiplier formulation are investigated. We give three typical thick brane background solutions with different asymptotic behaviors and show that all the solutions are stable under tensor perturbations. The effective potentials of the tensor perturbations exhibit as volcano potential, Poöschl–Teller potential, and harmonic oscillator potential for the three background solutions, respectively. All the tensor zero modes (massless gravitons) of the three cases can be localized on the brane. We also calculate the corrections to the four-dimensional Newtonian potential. On a large scale, the corrections to the four-dimensional Newtonian potential can be ignored. While on a small scale, the correction from the volcano-like potential is more pronounced than the other two cases. Combining the specific corrections to the four-dimensional Newtonian potential of these three cases and the latest results of short-range gravity experiments, we get the constraint on the scale parameter as $$k > rsim 10^{-4}$$ $k\gtrsim {10}^{-4}$ eV, and constraint on the corresponding five-dimensional fundamental scale as $$M_* > rsim 10^5$$ $M_{\ast }\gtrsim {10}^5$ TeV.