The abundant population of little red dots (LRDs), compact objects with red UV to optical colors and broad Balmer lines at high redshift, is revealing new insights into the properties of early active galactic nuclei (AGN). Perhaps the most surprising features of this population are the presence of Balmer absorption and ubiquitous strong Balmer breaks. Recent models link these features to an active supermassive black hole (SMBH) cocooned in very dense gas (NH ∼ 1024 cm−2). We present a stringent test of such models using VLT/MUSE observations of A2744-45924, the most luminous LRD known to date (LHα ≈ 1044 erg s−1), located behind the Abell-2744 lensing cluster at z = 4.464 (μ = 1.8). We detect a moderately extended Lyα nebula (h ≈ 5.7 pkpc), spatially offset from the point-like Hα seen by JWST by ≈1.6 pkpc. The Lyα emission is narrow (FWHM = 270 ± 15 km s−1), and faint (Lyα = 0.07Hα) compared to Lyα nebulae typically observed around quasars of similar luminosity. We detect compact N IV]λ1486 emission, spatially aligned with Hα, and a spatial shift in the far-UV continuum matching the Lyα offset. We discuss that Hα and Lyα have distinct physical origins: Hα originates from the AGN, while Lyα is powered by star formation. In the environment of A2744-45924, we identified four extended Lyα halos (Δz < 0.02, Δr < 100 pkpc). Their Lyα luminosities match the expectations based on Hα emission, and show no evidence for radiation from A2744-45924 affecting its surroundings. The lack of strong, compact, and broad Lyα and the absence of a luminous extended halo, suggest that the UV AGN light is obscured by dense gas cloaking the SMBH with a covering factor close to unity.