Does fading memory mean impulse response with finite support?
What you want is theoretically impossible: if $h(t) \neq 0$ only on a finite interval, then it is not bandlimited. Fortunately, all real-world signals and systems have the property that $H(f) \rightarrow 0$ as $|f| \rightarrow \infty$. This allows you to approximate $h(t)$ very closely with only a finite number of samples. For engineering purposes, this is often more than enough.
Note that the property above means that $h(t)$ has finite energy and is absolutely integrable (it is in $L_1$). For the math to work out, it is often also required that $h(t)$ is in $L_2$, or that its square is absolutely integrable. These are the terms that are commonly used to describe these kinds of signals and systems.