Is it possible to see the oscillations with plasma frequency in a gas of particles of the same charge (not mixture of positive and negative charges)?
I am not an expert on non-neutral plasma but I am going to explain what I know. One of the wave modes that exists in non-neutral plasma is Langmuir waves, which has a dispersion relation of (in neutral plasma):
The previous equation is also called Bohm-Gross dispersion relation. The terms in the right hand side are the plasma frequency of electrons squared and the other term is 3 by the wave vector squared by the thermal velocity of electrons squared. The previous equation allows for a zero value of the wave vector. In such a case the frequency of oscillation is equal to plasma frequency. In that case that oscillation is called plasma oscillation but it is not a wave since the wave vector is zero.
For non-neutral plasma, the plasma frequency as a parameter exists. Langmuir waves exist as well. If your question is whether plasma oscillations exist (for zero wave vector), then the answer I suppose is no. In this following papers have a look at the dispersion relation of Langmuir waves for purely ionic plasma and purely electronic plasma; you see that the frequency is zero when the wave vector is zero (I am not certain whether it is exactly zero or if it’s lower by orders of magnitude than the frequency ranges plotted). Comparing those plots with the same plots for Langmuir waves in neutral plasma, it is clear to see from the last equation that for zero wave vectors, the frequency of oscillation is non-zero. See the figure below for neutral plasma (taken from Chen chapter 4)
Hope that helped!!