Estimation with longitudinal data having nonignorable dropout is considered when the joint distribution of the study variable y and covariate x is nonparametric and the dropout propensity conditional on y and x follows a parametric model. To deal with the identifiability problem caused by nonignorable dropout, we use an instrument z, which is part of x related to y but unrelated to the dropout propensity conditioned on y and other covariates. We then apply the generalized method of moments to estimate the parameters in the dropout propensity model based on some estimating equations constructed using the instrument. Population parameters in the nonparametric distribution of y or conditional distribution of y given x can be estimated based on inverse propensity weighting. To improve efficiency, we derive a model-assisted regression estimator that is asymptotically more efficient, when the regression model is correct, and is robust against misspecification of the regression model. Consistency and asymptotic normality of the proposed estimators are established. The finite-sample performance of the estimators is studied through simulation, and an application to the HIV-CD4 data set is also presented as illustration.