Let $$ X\,\ :=\,\ [1;2)\ =\ \{x\in\mathbb R: 1\le x< 2)\} $$ The distance in $\ X\ $ is defined by:

$$ \forall_{x\ y\in X}\quad d(x\ y)\ :=\ \max(x\ y) $$

This ultra-metric space is discrete hence complete.

Also:

$$ \forall_{x\in X}\quad \{d(x\ y) : y\in X\}\ =\ [x;2)\ $$

Finally, the following descending sequence of nonempty closed balls $\ B(x\ r):=\{y\in X: d(x\ y)\le r\}\ $ has an empty intersection:

$$ \bigcap_{n=1}^\infty\ B\left(\frac{n+1}n\ \ \frac1n\right)\,\ =\,\ \emptyset $$ Thus, all four assumptions are satisfied.

Let $$ X\,\ :=\,\ (1;2)\ =\ \{x\in\mathbb R: 1< x< 2\} $$ The distance in $\ X\ $ is defined by:

$$ \forall_{x\ y\in X}\quad d(x\ y)\ :=\ \max(x\ y) $$

This ultra-metric space is discrete hence complete.

Also:

$$ \forall_{x\in X}\quad \{d(x\ y) : y\in X\}\ =\ [x;2)\ $$

Finally, the following descending sequence of nonempty closed balls $\ B(x\ r):=\{y\in X: d(x\ y)\le r\}\ $ has an empty intersection:

$$ \bigcap_{n=1}^\infty\ B\left(\frac{n+1}n\ \ \frac1n\right)\,\ =\,\ \emptyset $$ Thus, all four assumptions are satisfied.

Let $$ X\,\ :=\,\ [1;2)\ =\ \{x\in\mathbb R: 1\le x< 2)\} $$ The distance in $\ X\ $ is defined by:

$$ \forall_{x\ y\in X}\quad d(x\ y)\ :=\ \max(x\ y) $$

This ultra-metric space is discrete hence complete.

Also:

$$ \forall_{x\in X}\quad \{d(x\ y) : y\in X\}\ =\ [x;2)\ $$

Thus, all three assumptions are satisfied.

Let $$ X\,\ :=\,\ [1;2)\ =\ \{x\in\mathbb R: 1\le x< 2)\} $$ The distance in $\ X\ $ is defined by:

$$ \forall_{x\ y\in X}\quad d(x\ y)\ :=\ \max(x\ y) $$

This ultra-metric space is discrete hence complete.

Also:

$$ \forall_{x\in X}\quad \{d(x\ y) : y\in X\}\ =\ [x;2)\ $$

Finally, the following descending sequence of nonempty closed balls $\ B(x\ r):=\{y\in X: d(x\ y)\le r\}\ $ has an empty intersection:

$$ \bigcap_{n=1}^\infty\ B\left(\frac{n+1}n\ \ \frac1n\right)\,\ =\,\ \emptyset $$ Thus, all four assumptions are satisfied.