Thanks especially to Neil Strickland for the very helpful details.

Yes I thought I had put this detail in but it got lost at some point. The expected result says $\mathbb F_p[[x]]^\times\cong\mathbb F_p^\times\times\prod_{\mathbb N}\mathbb Z_p$ and in this notation $\mathbb N$ means nothing more than `a countable infinity'. I am interested in understanding the proof - yes - but I am sure this must be a standard result in a text book which is why I badged this as reference-request.

@JasonDeVito I think it likely that for a given perfect (finite) holonomy group $\phi$ there be more than one flat manifold with perfect fundamental group and with that holonomy, maybe of different dimensions. That maybe suggests that you would not expect the inequality $\dim N\le \dim M$. Or am I still missing some key point?

@JasonDeVito You are right that the holonomy group determines neither the fundamental group nor the first homology of the flat manifold. But it is always true that the abelianisation of the fundamental group is isomorphic to the first homology of the manifold.

@JasonDeVito Flat manifolds are aspherical which means that the homology of the manifold and its fundamental group coincide. On the group theory side, the fist homology of a group with trivial integer coefficients is isomorphic to the largest abelian quotient of the group. So perfect fundamental group amounts to the same condition as trivial first homology for these manifolds. In particular, it's not really a feature that is influenced by minimality.

@KasperAnderson J. A. Hillman emailed me to make some comments one of which was that the rank should be 15 not 12. I have gone over Plesken's formula applying Theorem V.1 when p=5 and I agree with Hillman: the minimal dimension with A_5 as holonomy seems to come out as 15 if we accept Plesken's formula. (I have not gone through the detail of Plesken's proof, I am just using the displayed formula on page 483 of his paper.

@KasperAnderson This is definitely a strong hint at 12. It seems implausible that allowing other perfect holonomy groups in place of $A_5$ would enable a reduction to 11 or less. I am just holding back from ticking this answer while I think about this last ingredient. I accept that at a common sense level, Plesken has addressed the question head on and has answers. The Plesken paper is quite technical and in some ways I feel it would be attractive to have a clean simple route into this question.

@CarloBeenakker If you simply follow through the calculation in Massey's book the minus sign I claim appears to be there. So at first I thought the first displayed formula above was a typo. But when its ramifications percolated through the subsequent text I started to question if I had done the calculation correctly. Tyrone agrees with my calculation. The notes you cite maybe do not shed great light because they don't include the nuts and bolts of the proofs.

If there were no such example would this not suggest there must be a super fantastic extension of Smith theory?