Abstract Objective. Deep image prior (DIP) has recently attracted attention owing to its unsupervised positron emission tomography (PET) image reconstruction method, which does not require any prior training dataset. In this paper, we present the first attempt to implement an end-to-end DIP-based fully 3D PET image reconstruction method that incorporates a forward-projection model into a loss function. Approach. A practical implementation of a fully 3D PET image reconstruction could not be performed at present because of a graphics processing unit memory limitation. Consequently, we modify the DIP optimization to a block iteration and sequential learning of an ordered sequence of block sinograms. Furthermore, the relative difference penalty (RDP) term is added to the loss function to enhance the quantitative accuracy of the PET image. Main results. We evaluated our proposed method using Monte Carlo simulation with [18F]FDG PET data of a human brain and a preclinical study on monkey-brain [18F]FDG PET data. The proposed method was compared with the maximum-likelihood expectation maximization (EM), maximum a posteriori EM with RDP, and hybrid DIP-based PET reconstruction methods. The simulation results showed that, compared with other algorithms, the proposed method improved the PET image quality by reducing statistical noise and better preserved the contrast of brain structures and inserted tumors. In the preclinical experiment, finer structures and better contrast recovery were obtained with the proposed method. Significance. The results indicated that the proposed method could produce high-quality images without a prior training dataset. Thus, the proposed method could be a key enabling technology for the straightforward and practical implementation of end-to-end DIP-based fully 3D PET image reconstruction.