Can any two points in an open and connected subspace of a locally convex t.v.s. be connected by a continuously differentiable path?

Question

Can any two points in an open and connected subspace of a Hausdorff locally convex space be connected by a continuously differentiable path?

Some known related facts:

An open and connected subspace of a locally convex space is path-connected (1, 2).

In a connected smooth manifold every two points can be connected by a continuously differentiable path (3, 4).