The answer is yes!
One can use Cover's theorem to prove it.
Remark 1: by "linear in some high dimensional space" we assume here, for any classifier, an existence of an equivalent linear classifier in higher dimensions (i.e., a decision hyperplane that yeilds identical results to the original classifier for any possible input).
Remark 2: it is definitely true for any classifier on a countable domain (e.g., all possible 8bit RGB 256x256 images is quite large but still a countable domain).
For classifiers on uncountable domains (like fields of real numbers), the answer would depend if Cover's theorem holds for infinitely large number of points.
However, I'd rather not be worried about that, since all real-world classifiers implemented on classical computers are discrete :-)
Remark 3: Of course, classifier must be deterministic...