Is it possible to get the function name in function body?

Try ${FUNCNAME[0]}. This array contains the current call stack. To quote the man page:

   FUNCNAME
          An  array  variable  containing the names of all shell functions
          currently in the execution call stack.  The element with index 0
          is the name of any currently-executing shell function.  The bot‐
          tom-most element is "main".  This variable exists  only  when  a
          shell  function  is  executing.  Assignments to FUNCNAME have no
          effect and return an error status.  If  FUNCNAME  is  unset,  it
          loses its special properties, even if it is subsequently reset.

The name of the function is in ${FUNCNAME[ 0 ]} FUNCNAME is an array containing all the names of the functions in the call stack, so:

$ ./sample
foo
bar
$ cat sample
#!/bin/bash

foo() {
        echo ${FUNCNAME[ 0 ]}  # prints 'foo'
        echo ${FUNCNAME[ 1 ]}  # prints 'bar'
}
bar() { foo; }
bar