Noncentrosymmetric superconductors can support flat bands of zero-energy surface states in part of their surface Brillouin zone. This requires that they obey time-reversal symmetry and have a sufficiently strong triplet-to-singlet-pairing ratio to exhibit nodal lines in the bulk. These bands are protected by a winding number that relies on chiral symmetry, which is realized as the product of time-reversal and particle-hole symmetry. We reveal a way to stabilize a flat band in the entire surface Brillouin zone, while the bulk dispersion is fully gapped. This idea could lead to a robust platform for quantum computation and represents an alternative route to strongly correlated flat bands in two dimensions, besides twisted bilayer graphene. The necessary ingredient is an additional spin-rotation symmetry that forces the direction of the spin-orbit-coupling vector not to depend on the momentum component normal to the surface. We define a winding number that leads to flat zero-energy surface bands due to bulk-boundary correspondence. We discuss under which conditions this winding number is nonzero in the entire surface Brillouin zone and verify the occurrence of zero-energy surface states by exact numerical diagonalization of the Bogoliubov-de Gennes Hamiltonian for a slab. In addition, we consider how a weak breaking of the additional symmetry affects the surface band, employing first-order perturbation theory and a quasiclassical approximation. We find that the surface states and the bulk gap persist for weak breaking of the additional symmetry but that the band does not remain perfectly flat. The broadening of the band strongly depends on the deviation of the spin-orbit-coupling vector from its unperturbed direction as well as on the spin-orbit-coupling strength and the triplet-pairing amplitude.