c. 1500 (implied in swampwatyr "swamp-water"), of uncertain origin, perhaps [Barnhart] a dialectal survival from an Old English cognate of Old Norse svöppr "sponge, fungus," from Proto-Germanic *swampuz; but traditionally connected with Middle English sompe "morass, swamp," which probably is from Middle Dutch somp or Middle Low German sump "swamp" (see sump). All of these likely are ultimately related to each other, from PIE *swombho- "spongy; mushroom," via the notion of "spongy ground."
[B]y swamps then in general is to be understood any low grounds subject to inundations, distinguished from marshes, in having a large growth of timber, and much underwood, canes, reeds, wythes, vines, briers, and such like, so matted together, that they are in a great measure impenetrable to man or beast .... [Bernard Romans, "A Concise History of East and West Florida," 1775]
More popular in U.S. (swamp (n.) by itself is first attested 1624 in Capt. John Smith's description of Virginia). Swamp-oak is from 1680s, American English. Swamp Yankee "rural, rustic New Englander" is attested from 1941. Thornton's "American Glossary" (1912) has swamp-angel "dweller in a swamp," swamp-law "might makes right."