In mixed near-field and far-field systems, the nonorthogonality between near-field and far-field channels may cause severe inter-user interference and hence degrade rate performance, when the analog beamforming is designed based on the low-complexity full-array maximum ratio transmission (MRT). To tackle this issue, we propose in this paper an antenna selection-based transmission framework to effectively suppress mixed-field interference without mechanically altering antenna structures. To this end, an optimization problem is formulated to maximize the sum-rate of mixed-field systems, by jointly designing antenna selection and power allocation under the MRT-based analog beamforming. As the problem is non-convex and generally difficult to solve optimally, we first consider a typical two-user scenario to obtain useful insights. Interestingly, we analytically show that the strong mixed-field interference can be substantially mitigated by deactivating only a small portion of antennas, yet without compromising array gains too much. Moreover, an inherent tradeoff is revealed in antenna selection between interference suppression and array-gain enhancement, based on which a suboptimal number of deactivated antennas for achieving the maximum sum-rate is obtained. Next, for the general multi-user case, we develop an efficient penalty dual decomposition (PDD)-based two-layer framework to obtain its high quality solution by using the block coordinate descent (BCD) and successive convex approximation (SCA) techniques. To further reduce the computational complexity, a low-complexity antenna deactivation strategy is proposed capitalizing on an interference suppression criterion. Last, numerical results demonstrate that the proposed scheme achieves a favorable trade-off between interference suppression and array gain loss, hence achieving significant performance gains over various baseline schemes.