diff --git a/lib/instrument/visitors/eval.js b/lib/instrument/visitors/eval.js
index 55d3d5c2..72e62437 100644
--- a/lib/instrument/visitors/eval.js
+++ b/lib/instrument/visitors/eval.js
@@ -165,14 +165,19 @@ function instrumentEvalCall(callNode, block, fn, isStrict, canUseSuper, superIsP
 			// If var is external to function, record function as using this var.
 			// Ignore `new.target` and `super` as they're not possible to recreate.
 			// https://github.com/overlookmotel/livepack/issues/448
-			if (externalVars && varName !== 'new.target' && varName !== 'super') {
-				activateBinding(binding, varName);
-				const externalVar = getOrCreateExternalVar(externalVars, block, varName, binding);
-				externalVar.isReadFrom = true;
-				if (!isConst) externalVar.isAssignedTo = true;
-				if (!externalVar.isFrozenName) {
-					externalVar.isFrozenName = true;
-					externalVar.trails.length = 0;
+			if (externalVars) {
+				if (varName === 'new.target' || varName === 'super') {
+					const externalVar = externalVars.get(block)?.[varName];
+					if (externalVar) externalVar.isFrozenName = true;
+				} else {
+					activateBinding(binding, varName);
+					const externalVar = getOrCreateExternalVar(externalVars, block, varName, binding);
+					externalVar.isReadFrom = true;
+					if (!isConst) externalVar.isAssignedTo = true;
+					if (!externalVar.isFrozenName) {
+						externalVar.isFrozenName = true;
+						externalVar.trails.length = 0;
+					}
 				}
 			}
 		}
diff --git a/lib/instrument/visitors/function.js b/lib/instrument/visitors/function.js
index 6885f766..38c6fa5f 100644
--- a/lib/instrument/visitors/function.js
+++ b/lib/instrument/visitors/function.js
@@ -762,6 +762,7 @@ function createFunctionInfoFunction(fn, state) {
 				return {
 					isReadFrom: varProps.isReadFrom || undefined,
 					isAssignedTo: varProps.isAssignedTo || undefined,
+					isFrozenName: varProps.isFrozenName || undefined,
 					isFrozenInternalName: varProps.binding.isFrozenName || undefined,
 					trails: varProps.trails
 				};
diff --git a/lib/serialize/blocks.js b/lib/serialize/blocks.js
index 4ea9fe97..483d559d 100644
--- a/lib/serialize/blocks.js
+++ b/lib/serialize/blocks.js
@@ -62,9 +62,11 @@ module.exports = {
 		const {containsEval} = block,
 			paramsByName = Object.create(null);
 		let frozenNamesIsCloned = false;
-		const params = [...block.params.keys()].map((name) => {
+		const params = [...block.params].map(([name, {isFrozenName}]) => {
+			if (containsEval) isFrozenName = true;
 			const param = {
 				name,
+				isFrozenName,
 				// Identifier nodes referring to this param
 				localVarNodes: [],
 				// If param always contains another function defined in this block,
@@ -83,7 +85,7 @@ module.exports = {
 			paramsByName[name] = param;
 
 			// `super` and `new.target` are not actually frozen
-			if (containsEval && !['super', 'new.target'].includes(name)) {
+			if (isFrozenName && !['super', 'new.target'].includes(name)) {
 				if (!frozenNamesIsCloned) {
 					frozenNames = new Set(frozenNames);
 					frozenNamesIsCloned = true;
@@ -637,7 +639,7 @@ module.exports = {
 
 			if (paramName === 'new.target') {
 				// `new.target` is always renamed as it can't be provided for `eval()` anyway
-				newName = transformVarName('newTarget', containsEval);
+				newName = transformVarName('newTarget', param.isFrozenName);
 
 				// Convert `MetaProperty` nodes into `Identifier`s with new name
 				for (const varNode of param.localVarNodes) {
@@ -665,7 +667,7 @@ module.exports = {
 			} else {
 				// Frozen var name (potentially used in `eval()`)
 				// eslint-disable-next-line no-lonely-if
-				if (paramName === 'this' || (paramName === 'arguments' && paramsByName.this)) {
+				if (paramName === 'this' || (paramName === 'arguments' && paramsByName.this?.isFrozenName)) {
 					// `this` or `arguments` captured from function.
 					// `arguments` is only injected with a function wrapper if `this` is frozen too.
 					// Otherwise, can just make `arguments` a normal param.
diff --git a/lib/serialize/functions.js b/lib/serialize/functions.js
index 99b95ba9..2593b3ef 100644
--- a/lib/serialize/functions.js
+++ b/lib/serialize/functions.js
@@ -150,12 +150,13 @@ module.exports = {
 				}
 
 				// Add var names to block
-				for (const [varName, {isAssignedTo}] of Object.entries(vars)) {
+				for (const [varName, {isAssignedTo, isFrozenName}] of Object.entries(vars)) {
 					const param = params.get(varName);
 					if (!param) {
-						params.set(varName, {isMutable: isAssignedTo || argNames?.has(varName)});
-					} else if (isAssignedTo) {
-						param.isMutable = true;
+						params.set(varName, {isMutable: isAssignedTo || argNames?.has(varName), isFrozenName});
+					} else {
+						if (isAssignedTo) param.isMutable = true;
+						if (isFrozenName) param.isFrozenName = true;
 					}
 				}
 
diff --git a/lib/serialize/parseFunction.js b/lib/serialize/parseFunction.js
index 6acc0a50..0f0a7fcd 100644
--- a/lib/serialize/parseFunction.js
+++ b/lib/serialize/parseFunction.js
@@ -82,7 +82,11 @@ module.exports = function parseFunction(
 			blockName,
 			vars: mapValues(vars, (varProps, varName) => {
 				externalVars[varName] = [];
-				return {isReadFrom: !!varProps.isReadFrom, isAssignedTo: !!varProps.isAssignedTo};
+				return {
+					isReadFrom: !!varProps.isReadFrom,
+					isAssignedTo: !!varProps.isAssignedTo,
+					isFrozenName: !!varProps.isFrozenName
+				};
 			})
 		});
 	}
@@ -867,14 +871,18 @@ function resolveFunctionInfo(
 		const {blockId} = scope;
 		if (blockId < fnId) {
 			// External var
-			for (const [varName, {isReadFrom, isAssignedTo, trails}] of Object.entries(scope.vars)) {
+			for (
+				const [varName, {isReadFrom, isAssignedTo, isFrozenName, trails}]
+				of Object.entries(scope.vars)
+			) {
 				if (isNestedFunction) {
 					const scopeDefVar = scopeDefs.get(blockId).vars[varName];
 					if (isReadFrom) scopeDefVar.isReadFrom = true;
 					if (isAssignedTo) scopeDefVar.isAssignedTo = true;
+					if (isFrozenName) scopeDefVar.isFrozenName = true;
 				}
 
-				externalVars[varName].push(...trailsToNodes(fnNode, trails, varName));
+				if (!isFrozenName) externalVars[varName].push(...trailsToNodes(fnNode, trails, varName));
 			}
 		} else {
 			// Var which is external to current function, but internal to function being serialized