We use LASSO methods to shrink, select, and estimate the high‐dimensional network linking the publicly traded subset of the world's top 150 banks, 2003–2014. We characterize static network connectedness using full‐sample estimation and dynamic network connectedness using rolling‐window estimation. Statically, we find that global bank equity connectedness has a strong geographic component, whereas country sovereign bond connectedness does not. Dynamically, we find that equity connectedness increases during crises, with clear peaks during the Great Financial Crisis and each wave of the subsequent European Debt Crisis, and with movements coming mostly from changes in cross‐country as opposed to within‐country bank linkages.