Yes, there is a possibility of undiscovered elements, even if they have very short lifetimes. The existence of such elements is based on theoretical predictions and the extrapolation of trends observed in known elements. These elements are often referred to as "superheavy elements" or "transactinide elements" and typically have atomic numbers higher than those of the currently known elements.
Superheavy elements are characterized by extremely unstable nuclei, which makes them difficult to produce and study. They often have very short half-lives, meaning they decay into lighter elements relatively quickly. The heaviest elements currently known are synthesized through nuclear reactions, usually by bombarding a target material with high-energy particles such as protons or nuclei of other elements.
To address the second part of your question, there are currently no known methods to prolong the lifetimes of superheavy elements significantly. The stability of an atomic nucleus depends on the balance between the strong nuclear force, which holds the protons and neutrons together, and the electrostatic repulsion between the positively charged protons. As the number of protons increases, the repulsive forces become stronger, making the nuclei less stable.
However, there have been efforts to explore potential ways to extend the lifetimes of superheavy elements. One approach is to investigate the properties of isotopes with specific combinations of protons and neutrons that may lead to more stable nuclei. Theoretical models and experimental studies aim to identify so-called "islands of stability" in the region of superheavy elements, where nuclei with relatively long lifetimes could exist.
While the synthesis and study of superheavy elements are challenging, advancements in experimental techniques and theoretical models continue to expand our understanding of these elements. Future discoveries of new elements and the exploration of their properties will contribute to our knowledge of nuclear physics and the understanding of the fundamental building blocks of matter.