No, not all members of the kingdom Protista possess flagella. The kingdom Protista is a diverse group of eukaryotic microorganisms that includes organisms such as protozoa, algae, and slime molds. While some protists, like many species of flagellates, do have flagella and use them for movement, there are numerous other protists that do not have flagella at all.
For instance, amoebae move by forming temporary projections called pseudopodia, rather than using flagella. Additionally, certain algae groups, like diatoms, primarily rely on other methods for locomotion or may be stationary. Thus, while flagella can be a common feature among some protists, it is not a universal characteristic of all members of the kingdom.