Spatial transcriptomic techniques can profile gene expressions while retaining the spatial information, thus offering unprecedented opportunities to explore the relationship between gene expression and spatial locations. The spatial relationship may vary across cell types, but there is a lack of statistical methods to identify cell-type-specific spatially variable (SV) genes by simultaneously modeling excess zeros and cell-type proportions. We develop a statistical approach CTSV to detect cell-type-specific SV genes. CTSV directly models spatial raw count data and considers zero-inflation as well as overdispersion using a zero-inflated negative binomial distribution. It then incorporates cell-type proportions and spatial effect functions in the zero-inflated negative binomial regression framework. The R package pscl is employed to fit the model. For robustness, a Cauchy combination rule is applied to integrate P-values from multiple choices of spatial effect functions. Simulation studies show that CTSV not only outperforms competing methods at the aggregated level but also achieves more power at the cell-type level. By analyzing pancreatic ductal adenocarcinoma spatial transcriptomic data, SV genes identified by CTSV reveal biological insights at the cell-type level. The R package of CTSV is available at https://bioconductor.org/packages/devel/bioc/html/CTSV.html. Supplementary data are available at Bioinformatics online.