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(Edited Feb 3 to correct previous wrong example)

Let me try a simpler example than in my comments. Consider the lattice $\mathbb{Z}^2$ and the polygon $P$ with vertices $(0,0)$, $(1,-3)$, $(2,-3)$, $(3,1)$, $(2,4)$ and $(1,4)$. (This is the Minkowski sum of vectors $(1,-3)$, $(1,0)$ and $(1,4)$). And consider $P$ itself as a "fundamental unit" for the lattice.

The points $(1,-1)$ and $(2,-1)$ are in $P$ but cannot be connected to the rest of lattice points in $P$ using the edges of $P$. (They can be connected to one another, but not to the rest). The same holds for the points $(1,2)$ and $(2,2)$.

(Edited Feb 3 to correct previous wrong example)

Let me try a simpler example than in my comments. Consider the lattice $\mathbb{Z}^2$ and the polygon $P$ with vertices $(0,0)$, $(1,-3)$, $(2,-3)$, $(3,1)$, $(2,4)$ and $(1,4)$. (This is the Minkowski sum of vectors $(1,-3)$, $(1,0)$ and $(1,4)$). And consider $P$ itself as a "fundamental unit" for the lattice.

The points $(1,-1)$ and $(2,-1)$ are in $P$ but cannot be connected to the rest of lattice points in $P$ using the edges of $P$. (They can be connected to one another, but not to the rest). The same holds for the points $(1,2)$ and $(2,2)$.

Update (Feb 4): this example is still wrong, but the one by Yoav Kallus in the comment below works.

(Edited Feb 3 to correct previous wrong example)

Let me try a simpler example than in my comments. Consider the lattice $\Z^2$ and the polygon $P$ with vertices $(0,0)$, $(1,-3)$, $(2,-3)$, $(3,1)$, $(2,4)$ and $(1,4)$. (This is the Minkowski sum of vectors $(1,-3)$, $(1,0)$ and $(1,4)$). And consider $P$ itself as a "fundamental unit" for the lattice.

The points $(1,-1)$ and $(2,-1)$ are in $P$ but cannot be connected to the rest of lattice points in $P$ using the edges of $P$. (They can be connected to one another, but not to the rest). Same for the points $(1,2)$ and $(2,2)$.

(Edited Feb 3 to correct previous wrong example)

Let me try a simpler example than in my comments. Consider the lattice $\mathbb{Z}^2$ and the polygon $P$ with vertices $(0,0)$, $(1,-3)$, $(2,-3)$, $(3,1)$, $(2,4)$ and $(1,4)$. (This is the Minkowski sum of vectors $(1,-3)$, $(1,0)$ and $(1,4)$). And consider $P$ itself as a "fundamental unit" for the lattice.

The points $(1,-1)$ and $(2,-1)$ are in $P$ but cannot be connected to the rest of lattice points in $P$ using the edges of $P$. (They can be connected to one another, but not to the rest). The same holds for the points $(1,2)$ and $(2,2)$.

Let me try a simpler example than in my comments. Consider the lattice $\Z^2$ and the polygon $P$ with vertices $(0,0)$, $(1,0)$, $(4,10)$, $(4,11)$, $(3,11)$ and $(0,1)$. And consider $P$ as a "fundamental unit" for the lattice.

The points of the form $(2,i)$ for $i\in \{4,5,6\}$ are in $P$ but cannot be connected to the rest of lattice points in $P$ using the edges of $P$.

(Edited Feb 3 to correct previous wrong example)

Let me try a simpler example than in my comments. Consider the lattice $\Z^2$ and the polygon $P$ with vertices $(0,0)$, $(1,-3)$, $(2,-3)$, $(3,1)$, $(2,4)$ and $(1,4)$. (This is the Minkowski sum of vectors $(1,-3)$, $(1,0)$ and $(1,4)$). And consider $P$ itself as a "fundamental unit" for the lattice.

The points $(1,-1)$ and $(2,-1)$ are in $P$ but cannot be connected to the rest of lattice points in $P$ using the edges of $P$. (They can be connected to one another, but not to the rest). Same for the points $(1,2)$ and $(2,2)$.