This fits perfectly the definition of a binomial distribution. There are $n$ independent trials, and in each trial the probability of "success," if you can call it success, is $p$. It is exactly like tossing a funny coin $n$ times, with the probability of a head equal to $p$, and our random variable the number of heads.
To put it another way, let $X_i=1$ if the $i$-th fabric is defective, and $0$ if it is not. Then
and the sum of $n$ independent identically distributed Bernoulli random variables has binomial distribution.
Remark: There is some connection with the Poisson. If $n$ is large and $p$ is small, with $np$ of modest size, then the binomial distribution of this problem is well-approximated by the Poisson with parameter $\lambda=np$. But it is still a binomial.