Panasonic LF1. I’ve just bought one and I’m incredibly impressed (note my other camera is a Canon 5DmkIII coupled to L series lenses so I have high expectations).
Very small (deck of cards) , can be full auto or fully manual or in between, actually has a viewfinder (albeit not a brilliant one) and can shoot raw if you want.
Quick to focus and to write to the memory card and 7x optical zoom.
Around £150 so a little above your budget but well worth it. When I was researching it it was one of the few p&s cameras I couldn’t find a bad review of.
Have to some excellent low light handheld shots with it through to bright daylight and it’s (for the class of camera) faultless.