  The nature of circumstellar envelopes (CSE) around Cepheids is still a matter
of debate. The physical origin of their infrared (IR) excess could be either a
shell of ionized gas, or a dust envelope, or both. This study aims at
constraining the geometry and the IR excess of the environment of the
long-period Cepheid $\ell$ Car (P=35.5 days) at mid-IR wavelengths to
understand its physical nature. We first use photometric observations in
various bands and Spitzer Space Telescope spectroscopy to constrain the IR
excess of $\ell$ Car. Then, we analyze the VLTI/MATISSE measurements at a
specific phase of observation, in order to determine the flux contribution, the
size and shape of the environment of the star in the L band. We finally test
the hypothesis of a shell of ionized gas in order to model the IR excess. We
report the first detection in the L band of a centro-symmetric extended
emission around l Car, of about 1.7$R_\star$ in FWHM, producing an excess of
about 7.0\% in this band. In the N band, there is no clear evidence for dust
emission from VLTI/MATISSE correlated flux and Spitzer data. On the other side,
the modeled shell of ionized gas implies a more compact CSE
($1.13\pm0.02\,R_\star$) and fainter (IR excess of 1\% in the L band). We
provide new evidences for a compact CSE of $\ell$ Car and we demonstrate the
capabilities of VLTI/MATISSE for determining common properties of CSEs. While
the compact CSE of $\ell$ Car is probably of gaseous nature, the tested model
of a shell of ionized gas is not able to simultaneously reproduce the IR excess
and the interferometric observations. Further Galactic Cepheids observations
with VLTI/MATISSE are necessary for determining the properties of CSEs, which
may also depend on both the pulsation period and the evolutionary state of the
stars.
