We study a problem of resource extraction cast as a stochastic control problem where the depletion time of the resource is modeled by the hitting time for the controlled dynamics of a random (non-observable) threshold. Such a threshold may represent a tipping point, i.e., a critical level below which we expect a drastic disruption of the underlying source, leading to its extinction. Mathematically, this is formulated as a singular control problem with random time- horizon. The underlying stochastic source X is singularly controlled by the cumulative extraction and it is modeled as a time-homogeneous diffusion process subject to general boundary conditions. The random time horizon is modeled by the first time X drops below a random thresh- old, which is independent of the Brownian motion and distributed according to a cdf F . The problem is cast in a Markovian setting by introducing the running infimum of X as an additional state variable, which leads to a 2-dimensional singular control problem with infinite time-horizon. Under some assumptions on F , we are able to fully characterize the solution of the problem. That is, we show that the optimal strategy consists of extracting resources in such a way that X reflects along a given boundary, which is expressed as a function of the running infimum. Depending on the chosen distribution F , the precise characterization of this boundary requires either solving an auxiliary problem or applying the so-called maximality principle, borrowed from optimal stopping theory, for singular control.