Let X be a (finite dimensional) manifold. Consider smooth mapping space $$PX = C^\infty(I, X)$$ where I = [0,1] is the closed interval. Is this space paracompact? What if we fix a point x in X and consider the pointed path space, is this space paracompact?
What topology do you want on $C^\infty(I,X)$? One natural topology, the one of uniform convergence of all derivatives, is metrizable, so paracompact.