  Based on HARPS-N radial velocities (RVs) and TESS photometry, we present a
full characterisation of the planetary system orbiting the late G dwarf
TOI-561. After the identification of three transiting candidates by TESS, we
discovered two additional external planets from RV analysis. RVs cannot confirm
the outer TESS transiting candidate, which would also make the system
dynamically unstable. We demonstrate that the two transits initially associated
with this candidate are instead due to single transits of the two planets
discovered using RVs. The four planets orbiting TOI-561 include an ultra-short
period (USP) super-Earth (TOI-561 b) with period $P_{\rm b} = 0.45$ d, mass
$M_{\rm b} =1.59 \pm 0.36$ M$_\oplus$ and radius $R_{\rm b}=1.42 \pm 0.07$
R$_\oplus$, and three mini-Neptunes: TOI-561 c, with $P_{\rm c} = 10.78$ d,
$M_{\rm c} = 5.40 \pm 0.98$ M$_\oplus$, $R_{\rm c}= 2.88 \pm 0.09$ R$_\oplus$;
TOI-561 d, with $P_{\rm d} = 25.6$ d, $M_{\rm d} = 11.9 \pm 1.3$ M$_\oplus$,
$R_{\rm d} = 2.53 \pm 0.13$ R$_\oplus$; and TOI-561 e, with $P_{\rm e} = 77.2$
d, $M_{\rm e} = 16.0 \pm 2.3$ M$_\oplus$, $R_{\rm e} = 2.67 \pm 0.11$
R$_\oplus$. Having a density of $3.0 \pm 0.8$ g cm$^{-3}$, TOI-561 b is the
lowest density USP planet known to date. Our N-body simulations confirm the
stability of the system and predict a strong, anti-correlated, long-term
transit time variation signal between planets d and e. The unusual density of
the inner super-Earth and the dynamical interactions between the outer planets
make TOI-561 an interesting follow-up target.
