We consider a combined state and drift estimation problem for the linear\nstochastic heat equation. The infinite-dimensional Bayesian inference problem\nis formulated in terms of the Kalman-Bucy filter over an extended state space,\nand its long-time asymptotic properties are studied. Asymptotic posterior\ncontraction rates in the unknown drift function are the main contribution of\nthis paper. Such rates have been studied before for stationary non-parametric\nBayesian inverse problems, and here we demonstrate the consistency of our\ntime-dependent formulation with these previous results building upon scale\nseparation and a slow manifold approximation.\n