A practical problem often encountered with observed count data is the presence of excess zeroes. The problem of zero-inflation in count data can easily be handled by zero-inflated models, which are mixtures of a point mass at zero and a discrete distribution for the count data. In the presence of predictors, zero-inflated Poisson (ZIP) regression models are, perhaps, the most commonly used. However, the parametric ZIP regression model could sometimes be restrictive, especially with respect to the mixing proportions. Taking inspiration from some of the recent literature on semiparametric mixtures of regression models for flexible mixture modeling, we propose a semiparametric ZIP regression model. We present an "EM-like" algorithm for estimation and a summary of asymptotic properties of the estimators. The proposed model is then applied to a dataset involving counts of clandestine methamphetamine laboratories.