  HD 209458b was the first transiting planet discovered, and the first for
which an atmosphere, in particular NaI, was detected. With time, it has become
one of the most frequently studied planets, with a large diversity of
atmospheric studies using low- and high-resolution spectroscopy. Here, we
present transit spectroscopy observations of HD 209458b using the HARPS-N and
CARMENES spectrographs. We fit the Rossiter-McLaughlin effect by combining
radial velocity data from both instruments (nine transits in total), measuring
a projected spin-orbit angle of $-1.6\pm0.3~{\rm deg}$. We also present the
analysis of high-resolution transmission spectroscopy around the NaI region at
$590~{\rm nm}$, using a total of five transit observations. In contrast to
previous studies where atmospheric NaI absorption is detected, we find that for
all of the nights, whether individually or combined, the transmission spectra
can be explained by the combination of the centre-to-limb variation and the
Rossiter-McLaughlin effect. This is also observed in the time-evolution maps
and transmission light curves, but at lower signal-to-noise ratio. Other strong
lines such as H$\alpha$, CaII IRT, the MgI triplet region, and KI D1 are
analysed, and are also consistent with the modelled effects, without
considering any contribution from the exoplanet atmosphere. Thus, the
transmission spectrum reveals no detectableNaI absorption in HD 209458b. We
discuss how previous pioneering studies of this benchmark object may have
overlooked these effects. While for some star-planet systems these effects are
small, for other planetary atmospheres the results reported in the literature
may require revision.
