If the bounded operators $X\to Y$ form a Banach space in the operator norm, is $Y$ necessarily Banach?

Question

I have seen that if $Y$ is Banach, the set $B(X,Y)$ of bounded linear operators from $X$ to $Y$ is Banach in the operator norm. I was now wondering about the converse. Is it true? More precisely:

Suppose $B(X,Y)$ is a Banach space, where $X,Y$ are two normed space. Is $Y$ then necessarily Banach?

I do not really know where to start, neither in trying to prove, nor in trying to find a counterexample.