@Mohan Thank you, could you give some ideas for the proof? I have thought about those questions since long ago..

@LaurentMoret-Bailly Theorem $3.16$ in this good survey solves the problem, and your work shows Theorem $3.16$ implies $A$ is an excellent Henselian local ring, so basically that's all we can say. I didn't notice that strong arrpoximation property before, thank you.

"p-DIVISIBLE GROUPS OVER Z" is at iopscience.iop.org/article/10.1070/IM1977v011n05ABEH001752, the section $3$ is short and includes the result of irregular primes. I am a beginner at this area, as there are some citations for those two papers and I am unable to find an Erratum... I apologize if there is any offense, if you like I can remove the words about V. A. Abrashkin's works. I just want to know more about this area, in particular some references for more recent works.

May I ask for some references for the positive results? Thank you.

@YCor thank you! I am just a little curious, I will ask another question instead.