You can use something like this:

---
- hosts: all 
  gather_facts: no
  become: yes
  tasks:
    - name: install git package
      apt:
        name: git

    - name: Get updated files from git repository 
      git: 
        repo: "https://{{ githubuser | urlencode }}:{{ githubpassword | urlencode | replace ('/', '%2f') }}@github.com/privrepo.git"
        dest: /tmp

Note: {{ githubpassword | urlencode | replace ('/', '%2f') }} is used here to account for special characters. replace ('/', '%2f') is necessary because urlencode does not convert /.

Then execute the following playbook:

ansible-playbook -i hosts github.yml -e "githubuser=arbabname" -e "githubpassword=xxxxxxx"

Note: Make sure you put the credentials in ansible vaults or pass it secure way

Why not just set the credentials directly into the repo path?

- name: Clone Project A git repo
  git:
    repo: 'https://{{ gitlab_project_A_deploy_username }}:{{ gitlab_project_A_deploy_password }}@company_gitlab_server.com/USER/Project_A.git'
    dest: /some/dir/

Following the comment from β.εηοιτ.βε I found how to make this git statement work in ansible.

First this has been to create an specific ssh key to be used as a deployment key:

ssh-keygen -t ed25519 -C "ansible deploy key for gitlab"

With password and the password stored in the ansible-vault (as gitlab_deploy_key_passwd). The file is saved in the path of the roles in ansible.

In the gitlab project, one has to go the "Settings > Repository > Deploy Keys" and place the content of the "gitlab_deploy_key_ed25519.pub".

In ansible, it is necessary to have a set of tasks to copy the private key (password protecte) in the remote as well as configure ssh to use it when talk with gitlab.

- name: "gitlab deploy key"
  block:
  - name: ".ssh directory with the best rights"
    file:
      path: /home/{{ ansible_user_id }}/.ssh
      state: directory
      mode: "u=rwx,g=,o="
      owner: "{{ ansible_user_id }}"
      group: "{{ ansible_user_id }}"
  - name: "gitlab deploy key copy"
    copy:
      src: "../files/gitlab_deploy_key_ed25519"
      dest: "/home/{{ ansible_user_id }}/.ssh"
      mode: "u=rwx,g=,o="
      owner: "{{ ansible_user_id }}"
      group: "{{ ansible_user_id }}"
  - name: "gitlab deploy key config"
    blockinfile:
      path: "/home/{{ ansible_user_id }}/.ssh/config"
      block: |
        Host gitlab.com
                User git
                Hostname gitlab.com
                IdentityFile /home/{{ ansible_user_id }}/.ssh/gitlab_deploy_key_ed25519

With this a "git clone" task can be setup:

- name: "git clone"
  expect:
    chdir: "{{ sources_dir }}"
    command: "git clone [email protected]:(...)/repo.git repo.git"
    responses:
      passphrase: "{{ gitlab_deploy_key_passwd }}"

So then, the repo is present in the remote without storing any private information there (or at least store it somehow protected). It is read-only and even this read requires a password that is as save as the ansible-vault. (I haven't understand why in this case the expect worked with the passphrase but didn't with the username/password).

This clones the repo without initializing the submodules, but from here the git commands in further ansible tasks will have access to the necessary things to do that.