Hyperbolic plasmonic metamaterials provide numerous opportunities for designing unusual linear and nonlinear optical properties. In this work, second‐harmonic generation in a hyperbolic metamaterial due to a free‐electron nonlinear response of a plasmonic component of the metamaterial is studied. It is shown that owing to a rich modal structure of an anisotropic plasmonic metamaterial slab, the overlap of fundamental and second‐harmonic modes results in the broadband enhancement of radiated second‐harmonic intensity by up to 2 orders of magnitude for TM‐ and TE‐polarized fundamental light, compared to a smooth Au film under TM‐polarised illumination. Compared to the radiated second‐harmonic intensity from a bulk LiNbO3 nonlinear crystal of the same thickness, the SHG intensity from a metamaterial slab may be up to 2 orders of magnitude higher at the certain metamaterial resonances. The results open up possibilities to design tuneable frequency‐doubling integratable metamaterial with the goal to overcome limitations associated with classical phase matching conditions in thick nonlinear crystals.