Charmed baryons are expected to exhibit a rich spectrum of states. However, only a few of these states have been confirmed and there must be many more excited states need to be found. Being one of them, the charmed baryon $\Lambda_c(2880)^+$ was observed by CLEO collaboration, BaBar collaboration and Belle collaboration, respectively. Now, more studies should still be required to enrich our understanding of $\Lambda_c(2880)$. Exploring the production of $\Lambda_c(2880)$ is helpful to reveal the inner structure of $\Lambda_c(2880)$ as well as studying the decay behavior of $\Lambda_c(2880)$. Therefore, we calculate the total and differential cross sections of $p\bar{p}\to \Lambda_c^-\Lambda_c(2880)^+$ by the effective Lagrangian approach under a $D$ meson exchanged mechanism. The numerical results indicate that $p\bar{p}\to \Lambda_c^-\Lambda_c(2880)^+$ is a suitable process to explore the $\Lambda_c(2880)^+$ production at $\overline{P}$ANDA. Considering the designed luminosity of $\overline{P}$ANDA (2$\times$10$^{32}$ cm$^{-2}$s$^{-1}$) and an overall efficiency of 10%, we can estimate that there are about 10$^{8}$ $\Lambda_c(2880)^+$ events accumulated per day. Based on the theoretical caculation, we suggest future $\overline{P}$ANDA experiment to perform the search for the charmed baryon $\Lambda_c(2880)^+$. This experimental study can not only further confirm $\Lambda_c(2880)^+$ by different processes, but also provide more abundant information to $\Lambda_c(2880)^+$, which will be valuable to reveal the underlying structure of $\Lambda_c(2880)^+$. As a new scientific project HIAF($\underline{H}$igh $\underline{I}$ntensity Heavy-Ion $\underline{A}$ccelerator $\underline{F}$acility) is being planned to be built in the near future, the proton beam with intensity of 3.0$\times$10$^{12}$ and energy of 12 GeV/c will be achieved. The possibilities for more hadron physics research will be also discussed in the talk.