  LHS 1140 is an M dwarf known to host two known transiting planets at orbital
periods of 3.77 and 24.7 days. The external planet (LHS 1140 b) is a rocky
super-Earth that is located in the middle of the habitable zone of this
low-mass star, placing this system at the forefront of the habitable exoplanet
exploration. We further characterize this system by improving the physical and
orbital properties and search for additional planetary-mass components in the
system, also exploring the possibility of co-orbitals. We collected 113 new
radial velocity observations with ESPRESSO over a 1.5-year time span with an
average photon-noise precision of 1.07 m/s. We determine new masses with a
precision of 6% for LHS 1140 b ($6.48 \pm 0.46~M_{\oplus}$) and 9% for LHS 1140
c ($m_c=1.78 \pm 0.17~M_{\oplus}$), reducing by half the previously published
uncertainties. Although both planets have Earth-like bulk compositions, the
internal structure analysis suggests that LHS 1140 b might be iron-enriched. In
both cases, the water content is compatible to a maximum fraction of 10-12% in
mass, which is equivalent to a deep ocean layer of $779 \pm 650$ km for the
habitable-zone planet LHS 1140 b. Our results also provide evidence for a new
planet candidate in the system ($m_d= 4.8\pm1.1~M_{\oplus}$) on a ~78.9-day
orbital period, which is detected through three independent methods. The
analysis also allows us to discard other planets above 0.5 $M_{\oplus}$ for
periods shorter than 10 days and above 2 $M_{\oplus}$ for periods up to one
year. Finally, our analysis discards co-orbital planets of LHS 1140 b down to 1
$M_{\oplus}$. Indications for a possible co-orbital signal in LHS 1140 c are
detected in both radial velocity and photometric data, however. The new
characterization of the system make it a key target for atmospheric studies of
rocky worlds at different stellar irradiations
