Conflict between Arduino delay function and SoftwareSerial

Question

It seems that I have a conflict between SoftwareSerial and the delay function on my Arduino (GeekCreit bought on Banggood). I am trying to use SoftwareSerial to send AT commands to an ESP-01.

When I perform:

#include <SoftwareSerial.h>;

SoftwareSerial esp8266(8,9);

void setup() {
  Serial.begin(9600);
  while (!Serial) ;
  esp8266.begin(9600);
  esp8266.println("AT");
}

void loop() {
 if(esp8266.available()) {
    while(esp8266.available()) {
      Serial.print(esp8266.read());
    }  
  }
}

Everything works well, the AT command is sent and I receive the response from my ESP.

But when I add a delay before sending the AT command, nothing happens: no command sent, no answer from the ESP.

#include <SoftwareSerial.h>;

SoftwareSerial esp8266(8,9);

void setup() {
  Serial.begin(9600);
  while (!Serial) ;
  esp8266.begin(9600);
  delay(2000);
  esp8266.println("AT");
}

void loop() {
 if(esp8266.available()) {
    while(esp8266.available()) {
      Serial.print(esp8266.read());
    }  
  }
}

Am I doing something wrong, has someone experienced the same problem?

I have tried to use AltSoftSerial instead but I have the same issue with it.