Christopher Davis for informing us of several relevant references.

Results and Proofs
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For the rest of the paper, we assume that $F$ is a field and that $p$ is a fixed positive real number greater than one.

Define $| \; |_\lambda$ on $F[[Y]]$ by $| f(Y) |_\lambda = \frac{1}{p^{\lambda({\text{ord}}_Y(f))}}$ for all $f$ in $F[[Y]]$.

This is basically the $Y$-adic absolute value on $F[[Y]]$, re-scaled by the growth function $\lambda(x)$.

$(F[[Y]], | \; |_\lambda$) is a complete normed ring.

We defined $\lambda(0) = 0$ and $\lambda(\infty) = \infty$, so $|a|_\lambda = 0$ if, and only if, $a = 0$ (because $\lambda$ is strictly increasing), and $|c_0| = 1$ for all $c_0$ in $F^\times$. Suppose that $f$ and $g$ are elements of $F[[Y]]$, then ${\text{ord}}_Y(fg) = {\text{ord}}_Y(f) + {\text{ord}}_Y(g)$, and $\lambda({\text{ord}}_Y(f) + {\text{ord}}_Y(g)) \geq \lambda({\text{ord}}_Y(f)) + \lambda({\text{ord}}_Y(g))$. Thus, $$\begin{aligned}
 |fg|_\lambda &=& \frac{1}{p^{\lambda({\text{ord}}_Y(f) + {\text{ord}}_Y(g))}} \\
&\leq& \frac{1}{p^{\lambda({\text{ord}}_Y(f))+\lambda({\text{ord}}_Y(g))}} \\
&=& |f|_\lambda |g|_\lambda.\end{aligned}$$ Similarly, since ${\text{ord}}_Y(f+g) \geq \min\{{\text{ord}}_Y(f), {\text{ord}}_Y(g)\}$ $$\begin{aligned}
 |f+g|_\lambda &=& \frac{1}{p^{\lambda({\text{ord}}_Y(f+g))}} \\
&\leq& \frac{1}{p^{\lambda(\min\{{\text{ord}}_Y(f), {\text{ord}}_Y(g)\})}} \\
&=& \max\{|f|_\lambda, |g|_\lambda\}.\end{aligned}$$

To show completeness, if $\left(f^{(i)}\right)_{i=1}^\infty$ is a Cauchy sequence with respect to the standard $Y$-adic norm $|\;|$ on $F[[Y]]$, then $f^{(i)}$ converges to an element $f$ in $F[[Y]]$. Thus, $|f - f^{(i)}| 