Is strick separation of points always possible with the Gaussian kernel when using support vector machine

Question

If I have a set of points in $\Bbb R^d$ that I want to separate, I can try to use the support vector machine with the Gaussian kernel. Through the feature map associated to the Gaussian kernel the points get mapped into an infinite dimentional space, and then a hyperplane that separates them is found there. The question is: from the theorical point of view, is strick separation always possible in the infinite dimensional space where the points are mapped? I can't read this statement anywhere.