In the question here, the subject of "Analysis in Positive Characteristic" is mentioned. Looking at Wikipedia's local field, this is the final type of analysis in local fields to be developed following on from real, complex and p-adic analysis. ( Is there a snappy name for this? Charpos analysis ? ff/Fq analysis ? Carlitz analysis ?)

Question: What is the situation with regard to division rings - Is there a classification of local division rings? and for which ones has analysis been developed ?