  We report a comprehensive de Haas--van Alphen (dHvA) study of the
heavy-fermion material CeRhIn$_5$ in magnetic fields up to 70~T. Several dHvA
frequencies gradually emerge at high fields as a result of magnetic breakdown.
Among them is the thermodynamically important $\beta_1$ branch, which has not
been observed so far. Comparison of our angule-dependent dHvA spectra with
those of the non-$4f$ compound LaRhIn$_5$ and with band-structure calculations
evidences that the Ce $4f$ electrons in CeRhIn$_5$ remain localized over the
whole field range. This rules out any significant Fermi-surface reconstruction,
either at the suggested nematic phase transition at $B^{*}\approx$ 30~T or at
the putative quantum critical point at $B_c \simeq$ 50~T. Our results rather
demonstrate the robustness of the Fermi surface and the localized nature of the
4$f$ electrons inside and outside of the antiferromagnetic phase.
