We present the first deep X-ray observations of luminous fast blue optical transient (LFBOT) AT 2018cow at ∼3.7 yr since discovery, together with the reanalysis of the observation at δ
t ∼ 220 days. X-ray emission is significantly detected at a location consistent with AT 2018cow. The very soft X-ray spectrum and sustained luminosity are distinct from the spectral and temporal behavior of the LFBOT in the first ∼100 days and would possibly signal the emergence of a new emission component, although a robust association with AT 2018cow can only be claimed at δ
t ∼ 220 days, while at δ
t ∼ 1350 days contamination of the host galaxy cannot be excluded. We interpret these findings in the context of the late-time panchromatic emission from AT 2018cow, which includes the detection of persistent, slowly fading UV emission with ν
L
ν
≈ 1039 erg s−1. Similar to previous works (and in analogy with arguments for ultraluminous X-ray sources), these late-time observations are consistent with thin disks around intermediate-mass black holes (with M
• ≈ 103–104
M
☉) accreting at sub-Eddington rates. However, differently from previous studies, we find that smaller-mass black holes with M
• ≈ 10–100 M
☉ accreting at ≳the Eddington rate cannot be ruled out and provide a natural explanation for the inferred compact size (R
out ≈ 40 R
☉) of the accretion disk years after the optical flare. Most importantly, irrespective of the accretor mass, our study lends support to the hypothesis that LFBOTs are accretion-powered phenomena and that, specifically, LFBOTs constitute electromagnetic manifestations of super-Eddington accreting systems that evolve to ≲Eddington over a ≈100-day timescale.