It states here that this Sir John Chichester was the brother of Sir Arthur Chichester, a future Lod Deputy of Ireland, and that Sir John died in Devon, England on 31st March 1585 at the age of 36. However, the brother of the future Lord Deputy, was killed at Carrickfergus on 4th November 1597. Something is amiss.
According to "Tudor Place" (http://www.tudorplace.com.ar/COURTENAY2.htm#Gertrude%20COURTENAY1), Gertrude Courtney-Chichester had 15 children of which 6 were males. One was Arthur,, the future Lord Deputy of Ireland, and the others were John, Charles, Adrian (I assume a male), Edward, and John 'the younger" whom I assume, as was his older and deceased brother of the same name, was named for his father..
However, I believe that the second John may have been listed as another child by mistake. That would have to be researched.