I've heard this described – not least by quiz question setters – as "the motto of the RNLI".
It's not just quiz question setters though. Take this blog for example.
The RNLI itself doesn't make a big thing of its motto, but I did find this page on the Institution's own web site, where it refers to "With courage, nothing is impossible" as "the motto of the institution".
Seahaven Maritime Academy (a training centre in Newhaven, Sussex, for professional and amateur mariners and sailors) agrees, naming "With courage, nothing is impossible" as "the motto of the Royal National Lifeboat Institution".
It appears on the charity's memorial, which can be seen outside its headquarters in Poole, Dorset. In this context the RNLI refers to it as "the motto of Sir William Hillary" (the Institution's founder).
"Let not the deep swallow me up" does appear on the RNLI's medals, but I could find no reference published by the Institution itself to suggest that this is its motto.
© Haydn Thompson 2020