Is it possible to make an SR latch using XOR gates only?
Not if we follow the normal logic design assumptions that each input will be fed by either a constant value or by exactly one output.

The problem is to make a SR latch we need to be able to force the output of a gate to a particular state, XOR gates can't do that, whatever the value of input A is, the output still depends on the value of input B.

If our logic family allows for "wired or" or "wired and" then it is possible. At that point we can basically use our XOR gates as buffers and inverters and use the "wired or" or "wired and" to perform the actual logic operations.