Gaussian moat

Let $\mathbb{Z}[i]=\{a+bi : a,b\in\mathbb{Z}\}$ be the ring of Gaussian integers, with norm $N(a+bi)=a^{2}+b^{2}$ and Euclidean absolute value $|a+bi|=\sqrt{a^{2}+b^{2}}$. Call $\pi\in\mathbb{Z}[i]$ a Gaussian prime if $\pi$ is a nonzero non-unit whose only divisors in $\mathbb{Z}[i]$ are units and unit multiples of $\pi$; equivalently, $\pi$ is a Gaussian prime exactly when $N(\pi)$ is a rational prime, or $\pi=u p$ with $u\in\{1,-1,i,-i\}$ and $p$ a rational prime with $p\equiv 3 \pmod 4$.

Then there is no infinite sequence $\pi_{1},\pi_{2},\pi_{3},\dots$ of pairwise distinct Gaussian primes such that

Equivalently, for each real $k>0$ let $G_{k}$ be the graph whose vertex set is the set of Gaussian primes and in which two distinct Gaussian primes $\pi,\pi'$ are joined by an edge when $|\pi-\pi'|\le k$; then every connected component of $G_{k}$ is finite. In particular, for every $k>0$ the component of $G_{k}$ containing the Gaussian prime $1+i$ is finite, so that only finitely many Gaussian primes can be reached from the origin by steps of length at most $k$ through Gaussian primes.