The text for the Manuals doesn't directly address this, but we can reasonably conclude the effect shouldn't be suppressed by connecting antimagic rules and the entire description of the Manual (in this case, Quickness of Action).
This book contains coordination and balance exercises, and its words are charged with magic. If you spend 48 hours over a period of 6 days or fewer studying the book's contents and practicing its guidelines, your Dexterity score increases by 2, as does your maximum for that score. The manual then loses its magic, but regains it in a century.
I've emphasized three key points in this paragraph- the first point is that the book's words are what is charged with magic.
The second point is that the magical effect of these words is to increase your Dexterity score and Dexterity maximum. This magical effect is the crux of your question- magical effects are normally suppressed inside an AM field by definition.
However, the third point is our resolution. After you gain the benefit of this magical effect of reading these words, the manual loses its magic, but you do not lose the ability increase, even if you lose the book. This means the magical effect is instantaneous and without duration.
If you hit someone with a fireball and then cover them with an antimagic field, they don't get the HP back, because the magic causing the effect has already passed.
The conclusion of this is that if you're reading the book in an antimagic field, you can't progress through or finish the 48 hours of reading, but once you've read the book, like any other instantaneous effect that modifies your stats, it sticks.
Finally, as a narrative point based on this text, you study the book and practice its guidelines. I believe a reasonable implication here is this manual's magic gives it the words that guide the person attuned to it how to best improve its current ability in themselves, kind of like a personal trainer/tutor in a book. Once you've completed this magically accelerated exercise/study, you are stronger, faster, smarter, etc, and while magic caused it, it's no longer magic.