diff --git a/clippy_lints/src/identity_op.rs b/clippy_lints/src/identity_op.rs
index b4e7bbc767135..f824f20ca40a0 100644
--- a/clippy_lints/src/identity_op.rs
+++ b/clippy_lints/src/identity_op.rs
@@ -61,15 +61,18 @@ impl<'tcx> LateLintPass<'tcx> for IdentityOp {
 }
 
 fn is_allowed(cx: &LateContext<'_>, cmp: BinOp, left: &Expr<'_>, right: &Expr<'_>) -> bool {
-    // `1 << 0` is a common pattern in bit manipulation code
-    cmp.node == BinOpKind::Shl
-        && constant_simple(cx, cx.typeck_results(), right) == Some(Constant::Int(0))
-        && constant_simple(cx, cx.typeck_results(), left) == Some(Constant::Int(1))
+    // This lint applies to integers
+    !cx.typeck_results().expr_ty(left).peel_refs().is_integral()
+        || !cx.typeck_results().expr_ty(right).peel_refs().is_integral()
+        // `1 << 0` is a common pattern in bit manipulation code
+        || (cmp.node == BinOpKind::Shl
+            && constant_simple(cx, cx.typeck_results(), right) == Some(Constant::Int(0))
+            && constant_simple(cx, cx.typeck_results(), left) == Some(Constant::Int(1)))
 }
 
 fn check(cx: &LateContext<'_>, e: &Expr<'_>, m: i8, span: Span, arg: Span) {
-    if let Some(Constant::Int(v)) = constant_simple(cx, cx.typeck_results(), e) {
-        let check = match *cx.typeck_results().expr_ty(e).kind() {
+    if let Some(Constant::Int(v)) = constant_simple(cx, cx.typeck_results(), e).map(Constant::peel_refs) {
+        let check = match *cx.typeck_results().expr_ty(e).peel_refs().kind() {
             ty::Int(ity) => unsext(cx.tcx, -1_i128, ity),
             ty::Uint(uty) => clip(cx.tcx, !0, uty),
             _ => return,
diff --git a/clippy_utils/src/consts.rs b/clippy_utils/src/consts.rs
index 04347672e0fbb..1cf8f9721b58f 100644
--- a/clippy_utils/src/consts.rs
+++ b/clippy_utils/src/consts.rs
@@ -168,6 +168,14 @@ impl Constant {
             None
         }
     }
+
+    #[must_use]
+    pub fn peel_refs(mut self) -> Self {
+        while let Constant::Ref(r) = self {
+            self = *r;
+        }
+        self
+    }
 }
 
 /// Parses a `LitKind` to a `Constant`.
diff --git a/tests/ui/identity_op.rs b/tests/ui/identity_op.rs
index ceaacaaf6bd70..2ed4b5db574d4 100644
--- a/tests/ui/identity_op.rs
+++ b/tests/ui/identity_op.rs
@@ -2,10 +2,20 @@ const ONE: i64 = 1;
 const NEG_ONE: i64 = -1;
 const ZERO: i64 = 0;
 
+struct A(String);
+
+impl std::ops::Shl<i32> for A {
+    type Output = A;
+    fn shl(mut self, other: i32) -> Self {
+        self.0.push_str(&format!("{}", other));
+        self
+    }
+}
 #[allow(
     clippy::eq_op,
     clippy::no_effect,
     clippy::unnecessary_operation,
+    clippy::op_ref,
     clippy::double_parens
 )]
 #[warn(clippy::identity_op)]
@@ -38,4 +48,9 @@ fn main() {
     42 << 0;
     1 >> 0;
     42 >> 0;
+    &x >> 0;
+    x >> &0;
+
+    let mut a = A("".into());
+    let b = a << 0; // no error: non-integer
 }
diff --git a/tests/ui/identity_op.stderr b/tests/ui/identity_op.stderr
index d8d44a74f9ab0..ff34b38db015b 100644
--- a/tests/ui/identity_op.stderr
+++ b/tests/ui/identity_op.stderr
@@ -1,5 +1,5 @@
 error: the operation is ineffective. Consider reducing it to `x`
-  --> $DIR/identity_op.rs:16:5
+  --> $DIR/identity_op.rs:26:5
    |
 LL |     x + 0;
    |     ^^^^^
@@ -7,64 +7,76 @@ LL |     x + 0;
    = note: `-D clippy::identity-op` implied by `-D warnings`
 
 error: the operation is ineffective. Consider reducing it to `x`
-  --> $DIR/identity_op.rs:17:5
+  --> $DIR/identity_op.rs:27:5
    |
 LL |     x + (1 - 1);
    |     ^^^^^^^^^^^
 
 error: the operation is ineffective. Consider reducing it to `x`
-  --> $DIR/identity_op.rs:19:5
+  --> $DIR/identity_op.rs:29:5
    |
 LL |     0 + x;
    |     ^^^^^
 
 error: the operation is ineffective. Consider reducing it to `x`
-  --> $DIR/identity_op.rs:22:5
+  --> $DIR/identity_op.rs:32:5
    |
 LL |     x | (0);
    |     ^^^^^^^
 
 error: the operation is ineffective. Consider reducing it to `x`
-  --> $DIR/identity_op.rs:25:5
+  --> $DIR/identity_op.rs:35:5
    |
 LL |     x * 1;
    |     ^^^^^
 
 error: the operation is ineffective. Consider reducing it to `x`
-  --> $DIR/identity_op.rs:26:5
+  --> $DIR/identity_op.rs:36:5
    |
 LL |     1 * x;
    |     ^^^^^
 
 error: the operation is ineffective. Consider reducing it to `x`
-  --> $DIR/identity_op.rs:32:5
+  --> $DIR/identity_op.rs:42:5
    |
 LL |     -1 & x;
    |     ^^^^^^
 
 error: the operation is ineffective. Consider reducing it to `u`
-  --> $DIR/identity_op.rs:35:5
+  --> $DIR/identity_op.rs:45:5
    |
 LL |     u & 255;
    |     ^^^^^^^
 
 error: the operation is ineffective. Consider reducing it to `42`
-  --> $DIR/identity_op.rs:38:5
+  --> $DIR/identity_op.rs:48:5
    |
 LL |     42 << 0;
    |     ^^^^^^^
 
 error: the operation is ineffective. Consider reducing it to `1`
-  --> $DIR/identity_op.rs:39:5
+  --> $DIR/identity_op.rs:49:5
    |
 LL |     1 >> 0;
    |     ^^^^^^
 
 error: the operation is ineffective. Consider reducing it to `42`
-  --> $DIR/identity_op.rs:40:5
+  --> $DIR/identity_op.rs:50:5
    |
 LL |     42 >> 0;
    |     ^^^^^^^
 
-error: aborting due to 11 previous errors
+error: the operation is ineffective. Consider reducing it to `&x`
+  --> $DIR/identity_op.rs:51:5
+   |
+LL |     &x >> 0;
+   |     ^^^^^^^
+
+error: the operation is ineffective. Consider reducing it to `x`
+  --> $DIR/identity_op.rs:52:5
+   |
+LL |     x >> &0;
+   |     ^^^^^^^
+
+error: aborting due to 13 previous errors