We solve a class of BSDE with a power function f(y) = y^q , q > 1, driving its drift and with the singular terminal boundary condition given by the indicator function of the ball B(m,r) or of its complement, where B(m, r) is the ball in the path space of continuous paths on [0,T] of the underlying Brownian motion centered at the constant function m and radius r. The solution involves the derivation and solution of a related heat equation in which f serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain the BSDE has continuous sample paths with the prescribed terminal value.