In this paper, a partial de-code-and-forward (P-DF) cooperative relaying network (CRN) with non-orthogonal multiple access (P-DF-CRN-NOMA) for maritime space communications is proposed. Without loss of generality, three kinds of relay forwarding protocols are studied to characterize the superiority of the proposed P-DF-CRN-NOMA scheme, where, after receptions from the coastal base station (BS), the island relay forwards the successful decoded symbols to the ship user to avoid the performance loss provided by the high decoding requirement in decode-and-forward (DF). To further improve the system performance, as well as for a fairness comparison, the maximum ratio combining (MRC) is utilized at the ship user. Specifically, to present the impact of the channel qualities, the scenarios of the weak and strong links that from the source to destination (S-D) are both considered. The closed-form expressions for the outage probability of the P-DF-CRN-NOMA are derived considering the three proposed forwarding cases. Qualitative numerical results corroborating our theoretical analysis, especially for the high signal-to-noise ratio (SNR), show that the proposed P-DF-CRN-NOMA scheme significantly improves the outage probability performance in comparison to the conventional CRN-NOMA schemes. Furthermore, for an increased S-D link, the corresponding outage probability performance will be also better for all the cases.