Yes, the ⟨p⟩ is always silent in words that start with ⟨ps⟩. The cluster /ps-/ is not a legit onset cluster in English because it violates the Phonotactics constraints of English.
Every language has a fixed set of rules called Phonotactics or Phonotactic rules/constraints that determine the permissible sequences of sounds. In simple words, Phonotactics studies what sounds go together, and where can they be found in a word.
For example, the sequence /pn-/ is allowed in Greek, but not in English. That's why the P in the word pneumonia is pronounced in Greek, but silent in English because English doesn't allow a cluster of PLOSIVE (/p t k/) + NASAL (/m n ŋ/) in the same syllable.
According to English Phonotactics constraints, English cannot have an onset cluster beginning with a plosive /p t k/ followed by a fricative /s z ʃ ʒ/. Therefore, the ⟨p⟩ in psychology and psychiatry is silent. Similarly, the ⟨t⟩ in the word tsunami is also silent in English.
Also note that the plosive is only silent when it's in the onset and precedes a fricative in the same syllable. In the syllable coda, however, the sequence 'Plosive + Fricative' is allowed.
/slashes/ represent phonemes (sounds) while ⟨angled brackets⟩ represent spelling.