"Can splish be used independently of splash? Can you splish without splash?" This random question was raised while we were driving about yesterday. Let's see what historical dictionaries say:
Neither splish nor splish-splash appear in Johnson's dictionary (1755), The Century Dictionary (1889-1891), or The New Century Dictionary (1940). In those, one can only splash.
In the 1st edition of the Oxford English Dictionary (1888-1928), splish-splash (and its derived form splishy-splashy) is present as a hyphenated, inseparable unit (volume 9/1, p. 645):
Splish-splash, v. rare. [f. Splash v.1, with usual variation of vowel.] intr. To splash repeatedly.
1720 Swift Irish Feast 44 The Floor is all wet... While the Water and Sweat, Splish, splash in their pumps. 1834 Medwin Angler in Wales I. 160 They went splish-splashing through an almost interminable inundation.
So Splishy-splashy a., sloppy, slushy. rare—1. c 1850 Denham Tracts (1895) II. 72 A cold, comfortless (splishy-splashy) Sabbath morning.
Similarly, splish-splash is presented succinctly as a single unit in Webster's Third New International Dictionary (1981) as both a verb ("to make a repeated splashing sound") and a noun ("a repeated splashing sound"), but there is no separate entry for splish alone (p. 2201).
So it would seem, based on these, that one cannot simply splish without the splash.