Dynamic positron emission tomography (PET) and kinetic modeling are pivotal in advancing tracer development research in small animal studies. Accurate kinetic modeling requires precise input function estimation, traditionally achieved via arterial blood sampling. However, arterial cannulation in small animals like mice, involves intricate, time-consuming, and terminal procedures, precluding longitudinal studies. This work proposes a non-invasive, fully convolutional deep learning-based approach (FC-DLIF) to predict input functions directly from PET imaging, potentially eliminating the need for blood sampling in dynamic small-animal PET. The proposed FC-DLIF model includes a spatial feature extractor acting on the volumetric time frames of the PET sequence, extracting spatial features. These are subsequently further processed in a temporal feature extractor that predicts the arterial input function. The proposed approach is trained and evaluated using images and arterial blood curves from [$^{18}$F]FDG data using cross validation. Further, the model applicability is evaluated on imaging data and arterial blood curves collected using two additional radiotracers ([$^{18}$F]FDOPA, and [$^{68}$Ga]PSMA). The model was further evaluated on data truncated and shifted in time, to simulate shorter, and shifted, PET scans. The proposed FC-DLIF model reliably predicts the arterial input function with respect to mean squared error and correlation. Furthermore, the FC-DLIF model is able to predict the arterial input function even from truncated and shifted samples. The model fails to predict the AIF from samples collected using different radiotracers, as these are not represented in the training data. Our deep learning-based input function offers a non-invasive and reliable alternative to arterial blood sampling, proving robust and flexible to temporal shifts and different scan durations.