Let $w:\mathbb{R}^n\rightarrow (0,1]$ be continuous, with $w(0)=1$ and $w(x)\rightarrow 0$ as $\|x\|\rightarrow \infty$.
Consider the vector space containing all continuous functions $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ with $\|f(x)\|w(x)\rightarrow 0$ as $\|x\|\rightarrow\infty$.
This space may be given the norm $\|f\|=\sup_{x\in\mathbb{R}^n}{\|f(x)\|w(x)}$.
Is this space complete?
(Intuitively it seems that it ought to be, as there is always some $x$ that attains the supremum in the norm, just like in the compact case. Proof: take the max over some compact neighbourhood of $0$. Then by convergence, there's some compact neighbourhood of the origin outside of which $\|f(x)\|w(x)$ is always less than this max. So now take the max over this new compact neighbourhood.)
Does this space possess a Schauder basis?
Consider the space $C_0(\mathbb{R}^{n})$ the space of function from $\mathbb{R}^{n}$ to $\mathbb{R}$ vanishing at infinity and equipped with the supremum norm. It is known to be complete and separable. We also denote by $C_{w}$ the space you are considering.
Now, set $T:C_0(\mathbb{R}^{n})\to C_{w}$ defined by $T(f)=\frac{f}{w}$ for all $f\in C_0(\mathbb{R}^{n})$.
This map is a (bijective) linear isometry, so $C_{w}$ inherits of all the (topological vector space) properties of $C_0(\mathbb{R}^{n})$.