In this paper we study local convergence of inexact Newton methods of the form (f(x(k)) + A(k)(x(k+1) - x(k)) + F(x(k+1)) boolean AND R-k(x(k)) not equal empty set with A(k) is an element of H(x(k)) for solving the generalized equation f(x) + F(x) (sic)0 in Banach spaces, where the function f is continuous but not necessarily smooth and F is a set-valued mapping with closed graph. The mapping H plays the role of a generalized set-valued derivative of f which in finite dimensions may be represented by Clarke's generalized Jacobian, while in Banach spaces it may be identified with Ioffe's strict prederivative. The set-valued mappings R-k represent inexactness. We utilize conditions divided into three groups: the first concerns the kind of nonsmoothness of the function f, the second involves metric regularity properties of an approximation of the mapping f + F, and the third is about the sequence of mappings R-k. Under various combinations of these conditions we show linear, superlinear, or quadratic convergence of the method. In the second part of the paper we give two generalizations of the Dennis-More theorem. As corollaries, we obtain results regarding convergence of inexact semismooth quasi-Newton-type methods and Dennis-More theorems for semismooth equations.