diff --git a/narwhals/_pandas_like/dataframe.py b/narwhals/_pandas_like/dataframe.py
index 796386fad..5eab8b08f 100644
--- a/narwhals/_pandas_like/dataframe.py
+++ b/narwhals/_pandas_like/dataframe.py
@@ -395,15 +395,13 @@ def with_columns(
         if not new_columns and len(self) == 0:
             return self
 
-        # If the inputs are all Expressions which return full columns
-        # (as opposed to scalars), we can use a fast path (concat, instead of assign).
+        # If the inputs are all Expressions
+        # (as opposed to Series), we can use a fast path (concat, instead of assign).
         # We can't use the fastpath if any input is not an expression (e.g.
         # if it's a Series) because then we might be changing its flags.
         # See `test_memmap` for an example of where this is necessary.
-        fast_path = (
-            all(len(s) > 1 for s in new_columns)
-            and all(isinstance(x, PandasLikeExpr) for x in exprs)
-            and all(isinstance(x, PandasLikeExpr) for (_, x) in named_exprs.items())
+        fast_path = all(isinstance(x, PandasLikeExpr) for x in exprs) and all(
+            isinstance(x, PandasLikeExpr) for (_, x) in named_exprs.items()
         )
 
         if fast_path:
diff --git a/narwhals/_pandas_like/utils.py b/narwhals/_pandas_like/utils.py
index 8ab2cfad7..87ed2c944 100644
--- a/narwhals/_pandas_like/utils.py
+++ b/narwhals/_pandas_like/utils.py
@@ -129,7 +129,7 @@ def validate_dataframe_comparand(index: Any, other: Any) -> Any:
         if other.len() == 1:
             # broadcast
             s = other._native_series
-            return s.__class__(s.iloc[0], index=index, dtype=s.dtype)
+            return s.__class__(s.iloc[0], index=index, dtype=s.dtype, name=s.name)
         if other._native_series.index is not index:
             return set_axis(
                 other._native_series,