The author derives existence conditions for nonlinear Hammerstein integral inclusions. First a multivalued continuation principle for upper semi-continuous, condensing maps is proved by using a fixed point theorem of E. Tarafdar and R. Výborný [Bull. Aust. Math. Soc. 12, 161-170 (1975; Zbl 0323.47044)]. This result is then used for deriving very general existence principles for the Hammerstein integral inclusion in a variety of settings. A general existence result is also given.