I would like to automate a huge amount of computation that involve basic arithmetic operations with $p$adic numbers. I have found a Mathematica packages for it, but it is old and acts quite erratically. Do you know of any computational software that does it reliably?

SAGE has padic arithmetic (for example, see http://www.math.utah.edu/~carlson/cimat/pythonsage.pdf), and has the added benefit of being completely free and opensource! 


This is just a very partial answer that is based on my experience trying to do some work with extension of padic numbers. There is padic arithmetic in the free software programs SAGE, PARI and GAP, but their main limitation (i don't know if things have changed recently) was their inability to with relative extension, that is extensions of another field which is itself a proper extension of $\mathbb{Q}_p$. However if you have limited need for extensions they (mostly SAGE and PARI, because GAP is more grouptheory oriented) have a very good interface to work with padic numbers. For my thesis work i absolutely needed relative extensions, so i had to use Magma which is not free, but for small computation it can be used online: http://magma.maths.usyd.edu.au/calc/. It has a very good library, which requires some time to learn, like the magma language which requires some learning too, but however the functionality provided is worth the effort. 

