Covariate-adaptive randomization (CAR) is widely used in clinical trials to balance treatment allocation over covariates. However, most results are established under the assumption that the covariates are correctly measured. In practice, misclassification is inevitable for discrete covariates. The impact of misclassification in covariates is twofold: it impairs the intended covariate balance, and raises concerns over the validity of test procedures. In this paper, we consider the impact of misclassification on covariate-adaptive randomized trials from the perspectives of both design and inference. We derive the asymptotic normality, and thereby the convergence rate, of the imbalance of the true covariates for a general family of CAR methods, and show that a superior covariate balance can still be attained compared to complete randomization. We also show that the two sample t-test is conservative, with a reduced Type I error. If the misclassified covariates are adjusted in the model used for analysis, the test maintains its nominal Type I error. Our results support the use of CAR in clinical trials, even when the covariates are subject to misclassification.