Infinite extent of spectrum, but also in time in Oppenheim's Discrete Time Signal Processing?
Not at all. The Uncertainty Principle says that a function cannot be both limited in time and limited in frequency. More specifically, the product of the signal's widths in time and in frequency (i.e., its time extension $\Delta_t$ and its bandwidth $\Delta_f$) is bounded from below:

$$\Delta_t\cdot\Delta_f\ge C\tag{1}$$

where the constant $C$ depends on the definition of bandwidth and time extension.

Note that $(1)$ is a lower bound, not an upper bound, so both widths can be infinite without contradicting $(1)$.

If a function is sharply localized in time then, by the Uncertainty Principle, it cannot be sharply localized in frequency, and vice versa. However, if - as in your example - a function is NOT localized in one of the two domains then this does not mean that it must be localized in the other domain. In may very well be non-localized in both domains, as is the case in the given example for non-integer $\alpha$.

Also take a look at this question and its answers for more details on the the Uncertainty Principle.