We study the subdifferential of \textit{F. H. Clarke} [Trans. Amer. Math. Soc. 205, 247-262 (1975; Zbl 0307.26012)] or random Lipschitz functions. We show that it is a multivalued mapping which is measurable in the collection of determinate and stochastic variables. The measurability of the Clarke subdifferential with respect to a random variable, which is necessary for its integration, is proved. A formula concerning the subdifferential calculus is proved in general. It is shown that the subdifferential expectation of a stochastic Lipschitz function is included in the expectation of the subdifferential of this function (as the integral of a multivalued map). Under assumptions similar to convexity, such an inclusion is established by \textit{L. Thibault} [C. R. Acad. Sci., Paris, Ser A 282, 507-510 (1976; Zbl 0343.46030)]. We note that for convex functions the Clarke subdifferential coincides with the ordinary subdifferential of a convex function (the set of subgradients). Thus, we have an inclusion of the subdifferential of the expectation of a stochastic convex function in the expectation of its subdifferential. The opposite inclusion follows trivially (under the measurability of the subdifferential which has been proved already) from the subgradient inequality for convex functions. Thus, in the convex case, we have a familiar result: the equality of the subdifferential of the expectation and the expectation of the subdifferential.