Arrow Keys Move Button

Indeed, you are correct that as soon as there is a focused button on the form, it will monopolize Key messages and Form1_KeyDown will no longer be called or move the picture box as you have observed.

The way around this is to implement a custom message filter by implementing IMessageFilter so that we can choose to intercept Win32 WM_KEYDOWN message unconditionally. Try something like this:

public partial class MainForm : Form, IMessageFilter
{
    public MainForm()
    {
        InitializeComponent();
        Application.AddMessageFilter(this);
        Disposed += (sender, e) =>Application.RemoveMessageFilter(this);
    }
    public bool PreFilterMessage(ref Message m)
    {
        if (m.Msg == WM_KEYDOWN)
        {
            var e = new KeyEventArgs(((Keys)m.WParam) | Control.ModifierKeys);
            OnHotKeyPreview(FromHandle(m.HWnd), e);
            return (e.Handled);
        }
        return false;
    }
    private void OnHotKeyPreview(Control? control, KeyEventArgs e)
    {
        switch (e.KeyData)
        {
            case Keys.Left: onMoveLeft(); e.Handled = true; break;
            case Keys.Up: onMoveUp(); e.Handled = true; break;
            case Keys.Right:  onMoveRight(); e.Handled = true; break;
            case Keys.Down: onMoveDown(); e.Handled = true; break;
        }
    }
    private void onMoveLeft() =>
        pictureBox1.Location = new Point(pictureBox1.Location.X - 15, pictureBox1.Location.Y);
    private void onMoveUp() => 
        pictureBox1.Location = new Point(pictureBox1.Location.X, pictureBox1.Location.Y - 15);
    private void onMoveRight() =>
        pictureBox1.Location = new Point(pictureBox1.Location.X + 15,
            pictureBox1.Location.Y);
    private void onMoveDown() =>
        pictureBox1.Location = new Point(pictureBox1.Location.X, pictureBox1.Location.Y + 15);
    private const int WM_KEYDOWN = 0x0100;
    private const int WM_HOTKEY = 0x0312;
}

---

Alternative

The cleaner approach, however, would be to inherit from PictureBox and implement the IMessageFilter there, to make an extended PictureBoxEx control that knows how to move itself by handling the intercepted key events.

To use this control, go to the Designer.cs file and swap out instances of PictureBox for extended class PictureBoxEx. Here's the final, alternative solution with the Form class simplified as a result. The scheme here is to set the border of the picture box(es) to move.

public partial class MainForm : Form
{
    public MainForm()
    {
        InitializeComponent();
        buttonDeselect.Click += (sender, e) =>
        {
            foreach (var pb in Controls.OfType<PictureBoxEx>())
            {
                pb.BorderStyle = BorderStyle.None;
            }
        };
    }
}

PictureBoxEx

class PictureBoxEx : PictureBox, IMessageFilter
{
    public PictureBoxEx()
    {
        Application.AddMessageFilter(this);
        Disposed += (sender, e) => Application.RemoveMessageFilter(this);
    }
    protected override void OnMouseDown(MouseEventArgs e)
    {
        base.OnMouseDown(e);
        if (ModifierKeys != Keys.Control)   // Allow multiple select.
        {
            foreach (var pb in Parent.Controls.OfType<PictureBoxEx>())
            {
                pb.BorderStyle = BorderStyle.None;
            }
        }
        BorderStyle = BorderStyle.FixedSingle;
    }
    private const int WM_KEYDOWN = 0x0100;
    public bool PreFilterMessage(ref Message m)
    {
        if ((m.Msg == WM_KEYDOWN) && (BorderStyle == BorderStyle.FixedSingle))
        {

            var e = new KeyEventArgs(((Keys)m.WParam) | Control.ModifierKeys);
            OnKeyDown(e);
            return (e.Handled);
        }
        return false;
    }
    protected override void OnKeyDown(KeyEventArgs e)
    {
        base.OnKeyDown(e);
        switch (e.KeyData)
        {
            case Keys.Left:
                Location = new Point(Location.X - 15, Location.Y);
                break;
            case Keys.Up:
                Location = new Point(Location.X, Location.Y - 15);
                break;
            case Keys.Right:
                Location = new Point(Location.X + 15, Location.Y);
                break;
            case Keys.Down:
                Location = new Point(Location.X, Location.Y + 15);
                break;
        }
    }
}