error: 'A' is an inaccessible base of 'B'

I've a code as follows -

#include <iostream>
#include <string>

class A{
    int a;
    public: virtual void sayHello(){ std::cout << "Hello\n"; }
};

class B : private A{
    std::string name;
  public:
    B(std::string _n): name(_n){}
    void sayName(){std::cout << name << "says hello\n";}
    void sayHello(){sayName();}
};


int main() {
    A *ptr = new B("c++");
    ptr->sayHello();
    return 0;
}

which produces the following compiler output -

Error:

prog.cpp: In function 'int main()':
prog.cpp:20:22: error: 'A' is an inaccessible base of 'B'
  A *ptr = new B("c++");
                      ^

As previously answered - here, here & here, I know how to solve this issue. By using public inheritence instead of private or protected.

But if I really really want to hide some interface behind the base class, isn't there some other way to do this? Or is it impossible to do so according to c++ lang specification.