Are all integers fractions?

Integers are fractions, because a number is itself no matter how you write it.

A relevant section from Lockhart's A Mathematician's Lament:

In place of a natural problem context in which students can make decisions about what they want their words to mean, and what notions they wish to codify, they are instead subjected to an endless sequence of unmotivated and a priori “definitions.” The curriculum is obsessed with jargon and nomenclature, seemingly for no other purpose than to provide teachers with something to test the students on. No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re equal for crying out loud. They are the same exact numbers, and have the same exact properties. Who uses such words outside of fourth grade?

Here's a relevant comic from SMBC:

I want to point out a different perspective. This is a typical issue where non-rigorous use of language leads to confusion. We often treat objects that are equivalent (under a possibly unspoken equivalence relation) as if they are equal.

Integers and rational numbers are not fractions, in the strictest sense of the word "are". For example, the fractions $1/1$, $4/4$, and $8/8$ are all different fractions, but they all represent the same integer. (Actually, I should say "the fractions represented by the expressions $1/1$, $4/4$, and $8/8$.") None of the fractions $1/1$, $4/4$, and $8/8$ literally is the integer 1. If they were all the same as the integer $1$, they would all be the same as each other - but they are not, because they are different fractions. What is true is that they have the same value as each other.

Similarly, the fractions $1/2$ and $2/4$ are not both the same as the rational number $1/2$, because they are not the same as each other, because they have different numerators. Who would claim that a rational number has a numerator?

When we write $$ 1 = 4/4 = 8/8 $$ the "equals" there only means that fractions represent the same number (they have the same value), not that they are the same fraction. In other words, the $=$ we write is actually an equivalence relation on a set of fractions - the equivalence relation being "has the same value". We have no symbol that we typically use to denote "actual" equality of fractions. This point is often ignored completely in lower-level texts.

The first place that people start to consider these things in detail is in abstract algebra. In that context, we run into many interesting distinctions that would not be visible in elementary math. For example, under the definitions in most algebra books, $\mathbb{Z}$ is an extension of the semigroup $\mathbb{N}$ to a group; $\mathbb{Q}$ is the field of fractions of $\mathbb{Z}$; and $\mathbb{R}$ is the Dedekind completion of $\mathbb{Q}$. It follows from these definitions that the natural number 1 is not the same as the integer 1, both are different than the rational number 1, and all three of those are different from the real number 1.

Every integer $x \in \mathbb Z$ can be expressed as the fraction $x \over 1$

I think you have a case to take to the teacher on this based on the teacher's use of the word "fraction" instead of "rational number". All integers are rational numbers also, everyone agrees (or should agree) on that. That is because all integers can be expressed as a ratio of integers. So here's the case:

Is $3$ a binary number?

I'd be surprised if too many people say "yes" to that question. Note that even though $3$ can be expressed as a binary number ($11$), we don't call it a "binary number" unless it is expressed as one.

Similarly, even though every integer can be expressed as a fraction (of integers, even), even mathematicians are likely to agree (for consistency's sake at least) that an integer is not itself a fraction even if it is always a rational number.