Bayesian Optimization (BO) is a surrogate-based global optimization strategy that relies on a Gaussian Process regression (GPR) model to approximate the objective function and an acquisition function to suggest candidate points. It is well-known that BO does not scale well for high-dimensional problems because the GPR model requires substantially more data points to achieve sufficient accuracy and acquisition optimization becomes computationally expensive in high dimensions. Several recent works aim at addressing these issues, e.g., methods that implement online variable selection or conduct the search on a lower-dimensional sub-manifold of the original search space. Advancing our previous work of PCA-BO that learns a linear sub-manifold, this paper proposes a novel kernel PCA-assisted BO (KPCA-BO) algorithm, which embeds a non-linear sub-manifold in the search space and performs BO on this sub-manifold. Intuitively, constructing the GPR model on a lower-dimensional sub-manifold helps improve the modeling accuracy without requiring much more data from the objective function. Also, our approach defines the acquisition function on the lower-dimensional sub-manifold, making the acquisition optimization more manageable. We compare the performance of KPCA-BO to a vanilla BO and to PCA-BO on the multi-modal problems of the COCO/BBOB benchmark suite. Empirical results show that KPCA-BO outperforms BO in terms of convergence speed on most test problems, and this benefit becomes more significant when the dimensionality increases. For the 60D functions, KPCA-BO achieves better results than PCA-BO for many test cases. Compared to the vanilla BO, it efficiently reduces the CPU time required to train the GPR model and to optimize the acquisition function compared to the vanilla BO.