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Unconventional superconductivity (USC) in a highly correlated kagome system has been theoretically proposed for years, yet the experimental realization is hard to achieve. The recently discovered vanadium-based kagome materials, which exhibit both superconductivity and charge density wave (CDW) orders, are nonmagnetic and weakly correlated, thus unlikely host USC as theories proposed. Here we report the discovery of a chromium-based kagome metal, CsCr$_3$Sb$_5$, which is contrastingly characterised by strong electron correlations, frustrated magnetism, and characteristic flat bands close to the Fermi level. Under ambient pressure, it undergoes a concurrent structural and magnetic phase transition at 55 K, accompanying with a stripe-like $4a_0$ structural modulation. At high pressure, the phase transition evolves into two transitions, probably associated with CDW and antiferromagnetic spin-density-wave orderings, respectively. These density-wave (DW)-like orders are gradually suppressed with pressure and, remarkably, a superconducting dome emerges at 3.65-8.0 GPa. The maximum of the superconducting transition temperature, $T_\mathrm{c}^{\mathrm{max}}=$ 6.4 K, appears when the DW-like orders are completely suppressed at 4.2 GPa, and the normal state exhibits a non-Fermi-liquid behaviour, reminiscent of USC and quantum criticality in iron-based superconductors. Our work offers an unprecedented platform for investigating possible USC in a correlated kagome system., Comment: 32 pages, 14 figures