Example of an integral domain which is not a field

For a counter-example, let's have a look at $\mathbb{Z} \subseteq \mathbb{Q}$.
Here $\mathbb{Z}$ is an integral domain which is not a field;
also you can check that $\mathbb{Z}$ is a sub-ring of the field of rational numbers $\mathbb{Q}$.

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Note that $\mathbb{Z}$ satisfies all of the field's properties; except the property which concerns the
existence of multiplicative inverses for non-zero elements. For example $2^{-1} \notin \mathbb{Z}$ .

No, subring of a field does not satisfy all the field's axioms. Namely, the problem is twofold: the subring doesn't have to contain $1$ and even when it does, there is trouble with inverses.

Let $1\in R\subseteq \mathbb F$, where $\mathbb F$ is a field and let $0\neq r\in R$. Sure, $r$ is invertible in $\mathbb F$, but what guarantees that $r^{-1}\in R$? Well, nothing. See the answer by Famke where they give example of $\mathbb Z\subseteq \mathbb Q$.

A yet another example are polynomials $R[x]$ over a field $R$. While (obviously) all other properties are fine, there are much less inverses than there are required for a field.

Now formal power series offer somewhat a remedy...