diff --git a/clippy_lints/src/unnecessary_sort_by.rs b/clippy_lints/src/unnecessary_sort_by.rs
index dd74bf367f3..26b56e0f2f3 100644
--- a/clippy_lints/src/unnecessary_sort_by.rs
+++ b/clippy_lints/src/unnecessary_sort_by.rs
@@ -1,6 +1,6 @@
 use clippy_utils::diagnostics::span_lint_and_sugg;
 use clippy_utils::sugg::Sugg;
-use clippy_utils::ty::is_type_diagnostic_item;
+use clippy_utils::ty::{implements_trait, is_type_diagnostic_item};
 use if_chain::if_chain;
 use rustc_errors::Applicability;
 use rustc_hir::{Expr, ExprKind, Mutability, Param, Pat, PatKind, Path, PathSegment, QPath};
@@ -193,10 +193,15 @@ fn detect_lint(cx: &LateContext<'_>, expr: &Expr<'_>) -> Option<LintTrigger> {
             let vec_name = Sugg::hir(cx, &args[0], "..").to_string();
             let unstable = name == "sort_unstable_by";
 
+            if_chain! {
             if let ExprKind::Path(QPath::Resolved(_, Path {
                 segments: [PathSegment { ident: left_name, .. }], ..
-            })) = &left_expr.kind {
-                if left_name == left_ident {
+            })) = &left_expr.kind;
+            if left_name == left_ident;
+            if cx.tcx.get_diagnostic_item(sym::Ord).map_or(false, |id| {
+                implements_trait(cx, cx.typeck_results().expr_ty(left_expr), id, &[])
+            });
+                then {
                     return Some(LintTrigger::Sort(SortDetection { vec_name, unstable }));
                 }
             }
diff --git a/tests/ui/unnecessary_sort_by.fixed b/tests/ui/unnecessary_sort_by.fixed
index b45b27d8f23..d806d620b17 100644
--- a/tests/ui/unnecessary_sort_by.fixed
+++ b/tests/ui/unnecessary_sort_by.fixed
@@ -2,6 +2,7 @@
 
 #![allow(clippy::stable_sort_primitive)]
 
+use std::cell::Ref;
 use std::cmp::Reverse;
 
 fn unnecessary_sort_by() {
@@ -33,6 +34,10 @@ fn unnecessary_sort_by() {
     // `Reverse(b)` would borrow in the following cases, don't lint
     vec.sort_by(|a, b| b.cmp(a));
     vec.sort_unstable_by(|a, b| b.cmp(a));
+
+    // No warning if element does not implement `Ord`
+    let mut vec: Vec<Ref<usize>> = Vec::new();
+    vec.sort_unstable_by(|a, b| a.cmp(b));
 }
 
 // Do not suggest returning a reference to the closure parameter of `Vec::sort_by_key`
diff --git a/tests/ui/unnecessary_sort_by.rs b/tests/ui/unnecessary_sort_by.rs
index be2abe7f701..6ee9c3af455 100644
--- a/tests/ui/unnecessary_sort_by.rs
+++ b/tests/ui/unnecessary_sort_by.rs
@@ -2,6 +2,7 @@
 
 #![allow(clippy::stable_sort_primitive)]
 
+use std::cell::Ref;
 use std::cmp::Reverse;
 
 fn unnecessary_sort_by() {
@@ -33,6 +34,10 @@ fn unnecessary_sort_by() {
     // `Reverse(b)` would borrow in the following cases, don't lint
     vec.sort_by(|a, b| b.cmp(a));
     vec.sort_unstable_by(|a, b| b.cmp(a));
+
+    // No warning if element does not implement `Ord`
+    let mut vec: Vec<Ref<usize>> = Vec::new();
+    vec.sort_unstable_by(|a, b| a.cmp(b));
 }
 
 // Do not suggest returning a reference to the closure parameter of `Vec::sort_by_key`
diff --git a/tests/ui/unnecessary_sort_by.stderr b/tests/ui/unnecessary_sort_by.stderr
index 50607933e18..ca9641e8803 100644
--- a/tests/ui/unnecessary_sort_by.stderr
+++ b/tests/ui/unnecessary_sort_by.stderr
@@ -1,5 +1,5 @@
 error: use Vec::sort here instead
-  --> $DIR/unnecessary_sort_by.rs:14:5
+  --> $DIR/unnecessary_sort_by.rs:15:5
    |
 LL |     vec.sort_by(|a, b| a.cmp(b));
    |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `vec.sort()`
@@ -7,67 +7,67 @@ LL |     vec.sort_by(|a, b| a.cmp(b));
    = note: `-D clippy::unnecessary-sort-by` implied by `-D warnings`
 
 error: use Vec::sort here instead
-  --> $DIR/unnecessary_sort_by.rs:15:5
+  --> $DIR/unnecessary_sort_by.rs:16:5
    |
 LL |     vec.sort_unstable_by(|a, b| a.cmp(b));
    |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `vec.sort_unstable()`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:16:5
+  --> $DIR/unnecessary_sort_by.rs:17:5
    |
 LL |     vec.sort_by(|a, b| (a + 5).abs().cmp(&(b + 5).abs()));
    |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `vec.sort_by_key(|a| (a + 5).abs())`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:17:5
+  --> $DIR/unnecessary_sort_by.rs:18:5
    |
 LL |     vec.sort_unstable_by(|a, b| id(-a).cmp(&id(-b)));
    |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `vec.sort_unstable_by_key(|a| id(-a))`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:20:5
+  --> $DIR/unnecessary_sort_by.rs:21:5
    |
 LL |     vec.sort_by(|a, b| (b + 5).abs().cmp(&(a + 5).abs()));
    |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `vec.sort_by_key(|b| Reverse((b + 5).abs()))`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:21:5
+  --> $DIR/unnecessary_sort_by.rs:22:5
    |
 LL |     vec.sort_unstable_by(|a, b| id(-b).cmp(&id(-a)));
    |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `vec.sort_unstable_by_key(|b| Reverse(id(-b)))`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:31:5
+  --> $DIR/unnecessary_sort_by.rs:32:5
    |
 LL |     vec.sort_by(|a, b| (***a).abs().cmp(&(***b).abs()));
    |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `vec.sort_by_key(|a| (***a).abs())`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:32:5
+  --> $DIR/unnecessary_sort_by.rs:33:5
    |
 LL |     vec.sort_unstable_by(|a, b| (***a).abs().cmp(&(***b).abs()));
    |     ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `vec.sort_unstable_by_key(|a| (***a).abs())`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:88:9
+  --> $DIR/unnecessary_sort_by.rs:93:9
    |
 LL |         args.sort_by(|a, b| a.name().cmp(&b.name()));
    |         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `args.sort_by_key(|a| a.name())`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:89:9
+  --> $DIR/unnecessary_sort_by.rs:94:9
    |
 LL |         args.sort_unstable_by(|a, b| a.name().cmp(&b.name()));
    |         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `args.sort_unstable_by_key(|a| a.name())`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:91:9
+  --> $DIR/unnecessary_sort_by.rs:96:9
    |
 LL |         args.sort_by(|a, b| b.name().cmp(&a.name()));
    |         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `args.sort_by_key(|b| Reverse(b.name()))`
 
 error: use Vec::sort_by_key here instead
-  --> $DIR/unnecessary_sort_by.rs:92:9
+  --> $DIR/unnecessary_sort_by.rs:97:9
    |
 LL |         args.sort_unstable_by(|a, b| b.name().cmp(&a.name()));
    |         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ help: try: `args.sort_unstable_by_key(|b| Reverse(b.name()))`