Let a and m be positive integers such that 2a < m. We show that in the domain $D:={z\in ℂ³|r\left(z\right):=\Re z₁+|z₁|²+|z₂|}^{2m}+{\left|z₂z₃\right|}^{2a}+{\left|z₃\right|}^{2m}<0$ the holomorphic sectional curvature $R_D(z;X)$ of the Bergman metric at z in direction X tends to -∞ when z tends to 0 non-tangentially, and the direction X is suitably chosen. It seems that an example with this feature has not been known so far.