The Proto-Indo-European root *peiH- "to be fat, swell" in the extended o-grade form *poid- became in Proto-Germanic *faitaz "fat". This became Old English fæt and then English fat.
In Proto-Celtic, the extended form *pī-wer- "fat, fertile" became *f–weryon- "earth, soil", which became Old Irish *īwer-iū "Ireland". This was borrowed into Old English as Īras "Irish", and thence our word Irish.
This is not completely certain; An etymological lexicon of Proto-Celtic notes in the entry for *f–weryon-:
"The Irish (and Welsh) name of Ireland, Ériu, W[elsh] Iwerddon, might also be related, but there are difficulties with this etymology, and there are alternative ones"
In Latin it became pītuīta "moisture exuded from trees." Thence English pituitary.
Another cognate is paneer, from Hindi पनीर panīr from Persian پنیر panīr from Indic *pēm "milk".
sources: Indogermanisches etymologisches Wörterbuch, An etymological lexicon of Proto-Celtic, The American Heritage Dictionary of Indo-European Roots 2001