Perhaps most relevant is [1] which discusses this construction and reduces the general case to $c=2$. See Theorem 1 and proposition 15 there. Note that $cp \le 1$ is not a sufficient condition; an inequality of the form $cp<1-\epsilon$ for some known $\epsilon>0$ is required. This follows from the criterion in the pioneering paper of Keane and O'Brien [2].
See also the interesting follow-up works [3], [4] that present some more efficient algorithms and implementations. In particular the introduction of [4] surveys the relevant literature up to 2016. In [5] it is shown that coins with success probability $cp$ for $c>1$ cannot be simulated in any interval of $p$ values via finite automata.

[1] Nacu, Şerban, and Yuval Peres. "Fast simulation of new coins from old." The Annals of Applied Probability 15, no. 1A (2005): 93-115.
https://projecteuclid.org/download/pdfview_1/euclid.aoap/1106922322

[2] Keane, M. S., and George L. O'Brien. "A Bernoulli factory." ACM Transactions on Modeling and Computer Simulation (TOMACS) 4, no. 2 (1994): 213-219.

[3] Łatuszyński, Krzysztof, Ioannis Kosmidis, Omiros Papaspiliopoulos, and Gareth O. Roberts. "Simulating events of unknown probabilities via reverse time martingales." Random Structures & Algorithms 38, no. 4 (2011): 441-452.

[4] Huber, Mark. "Nearly optimal Bernoulli factories for linear functions." Combinatorics, Probability and Computing 25, no. 4 (2016): 577-591.

[5] Mossel, Elchanan, and Yuval Peres. "New coins from old: computing with unknown bias." Combinatorica 25, no. 6 (2005): 707-724. https://arxiv.org/pdf/math/0304143.pdf