Context. Some models for the topology of the magnetic field in sunspot penumbrae predict regions free of magnetic fields or with only dynamically weak fields in the deep photosphere. Aims: We aim to confirm or refute the existence of weak-field regions in the deepest photospheric layers of the penumbra. Methods: We investigated the magnetic field at log τ5 = 0 is by inverting spectropolarimetric data of two different sunspots located very close to disk center with a spatial resolution of approximately 0.4-0.45''. The data have been recorded using the GRIS instrument attached to the 1.5-m solar telescope GREGOR at the El Teide observatory. The data include three Fe i lines around 1565 nm, whose sensitivity to the magnetic field peaks half a pressure scale height deeper than the sensitivity of the widely used Fe i spectral line pair at 630 nm. Before the inversion, the data were corrected for the effects of scattered light using a deconvolution method with several point spread functions. Results: At log τ5 = 0 we find no evidence of regions with dynamically weak (B< 500 Gauss) magnetic fields in sunspot penumbrae. This result is much more reliable than previous investigations made on Fe i lines at 630 nm. Moreover, the result is independent of the number of nodes employed in the inversion, is independent of the point spread function used to deconvolve the data, and does not depend on the amount of stray light (i.e., wide-angle scattered light) considered.