Say we have a normed field $K$ and that $X$ is a normed space over $K$.
The question has positive answer (i.e. discrete topology is not induced by any norm) if we implicitely assume that the normed field has a non-trivial norm. But if we further investigate into pathological cases, we can in fact find a normed space in which a trivial norm makes sense. So we have a counterexample.
Say $K$ is a finite field, then it can be shown that the only norm for $K$ is the trivial one. Now if we take the $K$-vector space $X = K^n$, then by fixing a basis (say, the canonical basis), there is a canonical way to define a norm over $X$, namely $ |(x_1, \dots, x_n)| := \max_{i = 1, \dots, n} |x_i|, $ and it turns out to be a trivial norm over $X$. So the induced topology is the discrete topology.
So, to be fair, we can say that the trivial topology is not induced by any norm if we assume that $K$ has a non-trivial topology (the argument is the same given by Brian M. Scott and by user38268).