I know in advance that, when writing a program in C or C++, even if I declare a function as "inline" the compiler is free to ignore this and decide not to expand it at each (or any) call.
Is the opposite true as well? That is, can a compiler automatically inline a very short function that wasn't defined as inline if the compiler believes doing so will lead to a performance gain?
Two other subquestions: is this behaviour defined somewhere in the ANSI standards? Is C different from C++ in this regard, or do they behave the same?
inline is non-binding with regards to whether or not a function will be inlined by the compiler. This was originally what it was intended to do. But since then, it's been realized that whether or not a function is worth inlining depends as much on the call site as the function itself and is best left to the compiler to decide.
Since this meaning of the keyword inline is non-binding, compilers are free to use inline substitution for any function that's not marked inline, and are free to generate function calls to any function marked inline. Those optimization choices do not change the rules regarding multiple definitions and shared statics listed above.
Edit : Since you asked for C as well, from https://en.cppreference.com/w/c/language/inline :
The intent of the inline specifier is to serve as a hint for the compiler to perform optimizations, such as function inlining, which require the definition of a function to be visible at the call site. The compilers can (and usually do) ignore presence or absence of the inline specifier for the purpose of optimization.