Suppose $k$ is a field and let $n > m$. Does there exist injective homomorphisms $$ k [[x_1, x_2, \ldots, x_n]] \rightarrow k[[x_1, x_2, \ldots, x_m]]\ ?$$
This answer is (a revision of) Simon Wadsley's comment posted here with the only goal to remove this question from the unanswered queue.
Let $k$ be a countable field, and take a map $k[[x,y,z]] \to k[[u,v]]$ that sends $x$ to $u$, $y$ to $uv$ and $z$ to $uf(v)$ for some $f∈k[[v]]$. It isn't hard to see that only for countably many choices of $f$ can the kernel possibly be non-zero.