Is it possible to have different genes in different parts of our body?
The general answer is that no, it's not possible to have different genes in different parts of the body.  There are two interesting classes of exceptions, but they are exceptions.

The first is that mutations can and do occur in cells during their growth.  Most of these are probably neutral - they occur in the vast wasteland of genetic junk in the chromosome, and nothing happens.  But some of them are harmful, and this is what puts cells on the pathway toward cancer.  Outright cancers, that we can detect, are the result of at least a half-dozen mutations occurring in the same cell lineage, that cumulatively allow the cells in that lineage to grow in uncontrolled manner, to avoid immune clearance, to encourage their own blood vessel growth, and so on.  (In fact, cancer cells generally have far more than a half-dozen mutations, because a route toward getting the mutations they need is to turn down DNA repair, allowing mutations to accumulate with minimal control.  For example, Genomic landscape of DNA repair genes in cancer, Genomic profiling of pediatric acute myeloid leukemia reveals a changing mutational landscape from disease diagnosis to relapse.)  

The second set of exceptions is in the immune system.  Both B cells and T cells specifically modify their chromosomes during their development, making huge numbers (trillions) or new genes.  The catch is that these are all in the same region, the antibody or T cell receptor genes.  It's part of the mechanism by which the adaptive immune system "remembers" which pathogens it's interacted with in the past.  The wikipedia article on V(D)J recombination is a starting point.