Green’s Dictionary of Slang

(orig. US) ‘nothing doing’, not a chance, no hope of that.

1918		Versailles Republican (IN) 3 Oct. n.p.: Saturday came along and we all dressed up in our best, as that was our liberty day, when the Commander came in and said ‘No Soap’ on liberty as we were in a draft. No one is allowed liberty when they are on a draft, afraid that someone would run away.
1924		P. Marks Plastic Age 99: If you don’t know, just say, ‘No soap,’ and if you do know, shoot your dope.
1939		J. Weidman What’s In It For Me? 115: But with that punk it was no soap.
1944		E.S. Gardner Case of the Crooked Candle (1958) 10: ‘No soap,’ Mason announced.
1956		J. Thompson ‘The Flaw in the System’ in Fireworks (1988) 134: I broke off, unable to tell him [...] that the deal was no soap.
1961		T.I. Rubin In the Life 24: I don’t go for the dyke stuff. Tried it a few times, but no soap.
1979		S. King Dead Zone (1980) 236: ‘No soap, huh?’ ‘No.’.
1982		M. Braun Judas Tree (1983) 42: ‘No soap,’ Starbuck said stubbornly.
1990		S. Morgan Homeboy 140: The Assistant D.A. is unwilling to cut any sort of deal. I tried to budge him, but no soap.
2004		J. Ellroy ‘Hot-Prowl Rape-O’ in Destination: Morgue! (2004) 312: I tried to get your phone number, but no fucking soap.