maximal compact subgroup as fixed points of some involution on p-adic group? As is well known, maximal compact subgroup of real Lie group is just the fixed points of Cartan involution. 
Now the question is what's the possible p-adic analog?
 A: There are some complications in the p-adic case. For GL_n, every maximal compact subgroup is conjugate to GL_n of the ring of integers, but for more general groups (eg PGL_2, SU_3) there can be more than one conjugacy class of maximal compact subgroups. One thing that you can say is that every maximal compact subgroup is the stabaliser of a point on the Bruhat-Tits building.
A: Here is how the real and p-adic situations are the same.
Let $G$ be a connected reductive algebraic group defined over a field $F$ not of characteristic two.  Let $\theta$ be an involution of $G$ defined over $F$.  Then the group $G^\theta$ of fixed points is a reductive algebraic subgroup of $G$.
Here are two ways in which they are different.
In the real case, one can always choose $\theta$ so that the group of rational points of $G^\theta$ is compact.  In the p-adic case, compact reductive groups are quite rare, and so in most cases there is no analogous way to choose $\theta$.
Second, compact subgroups do not play the same roles in the real and p-adic cases.  Think of the fields themselves.  In the p-adic case, the maximal compact subring is the ring of integers.  In the real case, there are no nontrivial compact subrings.   There is a ring of integers, but it is not compact.  Moreover, since $G^\theta$ has smaller dimension than $G$, it cannot be an open subgroup, and maximal compact subgroups are always open in the p-adic case.  Thus, even in the rare cases where $G^\theta$ is compact, it is not maximal.
