As we all know, metallic ions are surrounded by a sea of electrons. If we continually bend or stretch a metal, say, iron, it will break. Does that mean we break the metallic bond in subatomic level?
Yes, you are breaking metallic bonds. Similarly, when you crush a crystal of salt, you are breaking ionic bonds. There aren't that many good examples of breaking covalent bonds1, though breaking a polymer like nylon usually will do it.
The "physical" breaking of bonds is not a very exotic thing. Remember that while the energy required to break every bond in a gram of substance is a lot, the energy required to just break a slice of those bonds isn't much.
A very common question that arises is "what happens with the open valencies left behind when you physically break bonds?". Usually, gases from the atmosphere adsorb onto the metal surface, forming hydrogen bonds.
A note: When you do this, there is nothing going on at the subatmoic level, only at the atomic level.
1. This is not due to the strength of covalent bonds -- indeed, these are usually weaker than ionic bonds. This happens because most covalent substances are made up of smaller molecules (as opposed to macroscopic lattices) held together by hydrogen bonding/Van der Waals forces, and it is these bonds that break when you break a covalent substance.
In addition to ManishEarth’s answer, I'll add that this is only strictly true if you take a pure single crystal of metal and break it. If you take a real-life piece of iron, it is not a pure crystal but contains a large number of crystalline defects. The most common (and easiest to think about) in a polycrystalline material is the presence of grain boundaries between the many crystallites that make up the metal particle:
Micrograph of a polycrystalline metal; grain boundaries evidenced by acid etching (from Wikipedia)
So, your metal piece is not a single crystal, and crystalline defects are key in the understanding of mechanical response to applied stress, including the case of fracture in which you are interested. So, by breaking a piece of metal in two, you may not actually break so many metallic bonds :)