Pointed out on famous disbelief, I know now that there is an embedding

$\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$

for any $n \leq \infty$. Then I would like to ask

Q. Is any complete local sub-algebra $\iota_A \colon A \hookrightarrow {\Bbb F}_p[[X]]$ noetherian of Krull dimension $1$?

That is, I guess that complete local ring $A$ having an embedding $\iota \colon A \hookrightarrow {\Bbb F}_p[[X]]$ must be a noetherian local ring of Krull-dimension $1$.