$[\mathbb{N}]^n$, with edges between $u,v\in[\mathbb{N}]^n$ if $\#(u\cap v)=n-1$, is an infinite graph in which all vertices have infinite degree. Moreover, for any two vertices $u,v$ in $[\mathbb{N}]^n$ and any finite subset $E\subseteq[\mathbb{N}]^n$ not containing $a,b$, there is a path from $a$ to $b$ which does not pass through any vertices of $E$: to find it, one can consider some $N\in\mathbb{N}$ bigger than all natural numbers appearing in elements of $E$, and then you go from $a$ to $\{N+1,\dots,N+n\}$ and from $\{N+1,\dots,N+n\}$ to $b$ in $\leq 2n$ steps.

So one can create a Hamiltonian path in the following way:

• Number the vertices $u_1,u_2,\dots$ in $[\mathbb{N}]^n$.

• Start the path in the vertex $u_1$.

• At each step, take the first vertex that you have not visited and visit it using a path that does not pass through any previously visited vertices.

$[\mathbb{N}]^n$, with edges between $a,b\in[\mathbb{N}]^n$ if $\#(a\cap b)=n-1$, is an infinite graph in which all vertices have infinite degree. Moreover, for any two vertices $a,b$ in $[\mathbb{N}]^n$ and any finite subset $E\subseteq[\mathbb{N}]^n$ not containing $a,b$, there is a path from $a$ to $b$ which does not pass through any vertices of $E$: to find it, one can consider some $N\in\mathbb{N}$ bigger than all natural numbers appearing in elements of $E$, and then you go from $a$ to $\{N+1,\dots,N+n\}$ and from $\{N+1,\dots,N+n\}$ to $b$ in $\leq 2n$ steps.

So one can create a Hamiltonian path in the following way:

• Number the vertices $u_1,u_2,\dots$ in $[\mathbb{N}]^n$.

• Start the path in the vertex $u_1$.

• At each step, take the first vertex that you have not visited and visit it using a path that does not pass through any previously visited vertices.