If we take the Fourier transform of any constant signal, we get an
impulse at zero, which says that its frequency is zero and, hence, it
is non-repeating and its period is infinity.
No, this does not work for a zero signal (Fourier is flat-flat, no impulse). Plus, something over zero is traditionally undefined, and could be any number. And THIS is the very case here.
A periodic function $P$ is such that its values repeat "in some way" at "regular intervals or periods". Trigonometric functions are natural examples. Indeed, you can derive them from the exponential series, which is convergent and its own derivative
This notion of repetition can be multidimensional in the variable $x$. There are multiple avatars for periodicity: the most commonly used are simply periodic, doubly periodic, triply periodic functions.
In the complex plane (the most useful for signal practitioners so far), you can have a doubly periodic function $P$, with two incommensurate least/fundamental complex periods:
with $x_1/x_2$ is not real. Those are called elliptic functions. If you now restrict to univariate single-valued functions, then a theorem by Jacobi states that is is impossible for them to have more than two distinct (least/fundamental) periods.
But a periodic function can have "no least period" at all, and this is the case with constant functions:
The constant function $P(x)=0$ is periodic with any period $x_0$ for all
nonzero real numbers $x_0$, so there is no concept analogous to the least
period for constant functions.