Turns out that Gneiting and Raftery give an example in Section 4.2 of the continuous ranked probability score (CRPS), which is strictly proper for $\mathcal{P}$ equal to the Borel probability measures on $\mathbb{R}$. Writing a forecast as a cumulative distribution function $F$, the score is
$$ CRPS(F, x) = -\int_{-\infty}^{\infty} \Big(F(y) - \mathbf{1}[y \geq x] \Big)^2 dy . $$
It "corresponds to the integral of the Brier scores for the associated probability forecasts at all real-valued thresholds." Apparently this is a popular scoring rule in statistics.