  We investigate arithmetic aspects of the middle degree cohomology of
compactified Picard modular surfaces $X$ attached to the unitary similitude
group $\mathrm{GU}(2,1)$ for an imaginary quadratic extension $E/\mathbf{Q}$.
We construct new Beilinson--Flach classes on $X$ and compute their Archimedean
regulator. We obtain a special value formula involving a non-critical $L$-value
of the degree six standard $L$-function, a Whittaker period, and the regulator.
This provides evidence for Beilinson's conjecture in this setting.
