Answer: No. Just as in English, the initial p is silent. The same is true for the other words of Greek origin that begin with ps-, such as psicoanalizar (to psychoanalyze).
Some authorities recommend spelling such words without the initial p, and you'll often see them that way: sicología, siquiatra and sicoanalizar. Some words where the silent "p" is retained in English are universally spelled without the p in Spanish, among them salmo (psalm) and seudónimo (pseudonym).
Also, as in English, other words with initial silent letters include those of Greek origin that begin with pt- (such as pterodáctilo, pterodactyl), gn- (e.g., gnóstico, gnostic) and mn- (e.g., mnemotécnico, mnemonic). Most such words, few of which are common, can be spelled either with the initial silent letter or without.
The other silent letter you're most likely to come across is the n when it is followed by an m. This occurs most often in words beginning with inm- such as inmenso (immense) and inmunizar (to immunize). Also, when an m follows an n in adjoining words, as in un médico ("a doctor"), the sound of the n is often dropped.
One word that doesn't fit any of the above patterns is a word most commonly used in Mexico, tlapalería, which has an initial silent t. Meaning either a hardware store or a store where painting supplies are sold, it comes from an indigenous word for "color."
Additionally, of course, modern words of foreign origin often retain the original pronunciation and spelling (or roughly so) and thus can have silent letters as well.