White noise has the ACF: $R_{WW}[\kappa] = c_0 \delta [\kappa]$ and zero mean $m_W[\kappa] = 0$.
The first and second order moments of a WSS process depend only upon the time difference $\kappa$. With that being said, since $m_W $ is always zero, it does not depend on time and the ACF doesn't either which is (according to my book) sufficient to prove the WSS property of a process.
Am I right with this? Solutions to an exam I'm doing says that white noise must not always be WSS?
Well, this depends on your definition of white noise. This question asks for that definition.
One answer gives:
A white noise process is a random process of random variables that are uncorrelated, have mean zero, and a finite variance. Formally, $X(t)$ is a white noise process if $E(X(t))=0,E(X(t)^2)=S^2$, and $E(X(t)X(h))=0$ for $t≠h$. A slightly stronger condition is that they are independent from one another; this is an "independent white noise process."
Under this definition (the first of the two, the weaker one), which is I presume the same definition you have, your reasoning is perfectly correct and white noise is always wide-sense stationary. Note that this definition asks for the variances be finite, what I think you also do since you probably mean a finite number when you write $c_0$.
For the stronger version of the definition given in the same quoted answer, the same applies.
In contrast, the definition that Dilip Sarwate gave in his answer doesn't require the variances to be finite and hence allows for a white noise not to be wide-sense stationary as he explained.
There are probably other definitions for white noise out there. Possibly, in the context of your exam another definition of white noise is assumed than the one in your book and therefore the apparent contradiction.
White noise has the properties that you state, but those properties are not the properties that define white noise. As Michael Chernick's comment points out, a (discrete-time) white noise process is a collection of independent identically distributed zero-mean random variables, one for each time instant under consideration. If the random variables have finite variance $\sigma^2$, then the autocorrelation function of the process is $\sigma^2\delta[n]$ where $\delta[n]$ is the Kronecker delta function defined by $$\delta[n] = \begin{cases}1, &n=0,\\0, &n \neq 0.\end{cases}$$ Now, if the common distribution function of random variables does not have a variance, e.g. Cauchy random variables, then white noise is not a wide-sense-stationary process (even though it is a strictly stationary process).
Wise sense stationarity implies weak or covariance stationarity, i.e only the first two moments(mean and variance) are time-invariant or constant. A white noise time series in its simplest form has 0 mean, constant variance and is serially uncorrelated. Hence white noise implies wide-sense stationarity
